Define logarithms as indices, convert between index form and logarithmic form, apply the logarithm laws (product, quotient, power), use the logarithms of 1 and of the base, change the base, and work with common logarithms

What this dot point is asking

The previous page showed that a power $a^n$ is built from a base $a$ and an index $n$. A logarithm runs that idea backwards: it answers the question "to what power must I raise the base to get this number?". NESA expects you to define a logarithm as that index, to convert fluently between index form and log form, to use the three log laws (product, quotient and power) to expand and contract expressions, to know the logs of $1$ and of the base instantly, to change a logarithm from one base to another, and to handle common logarithms (base $10$), the ones your calculator does directly. The mechanics are short, but logarithms are the tool that finally lets you solve an exponential equation such as $5^x = 18$ exactly, which the index-matching method of the previous page could not reach. They also unlock the scales scientists use for quantities that span enormous ranges: pH for acidity, decibels for loudness, the Richter scale for earthquakes. The goal is to read a logarithm as an index without hesitation, so every law below becomes an old index law in a new costume.

The answer

A logarithm IS the index

Fix a base $a$ that is positive and not equal to $1$. The logarithm base $a$ of a positive number $x$ is the index you put on $a$ to produce $x$. In symbols,

Read $\log_2 8$ as "log base $2$ of $8$", saying the base first and then the number. Its value is $3$, because $3$ is the index when $8$ is written as a power of $2$: $8 = 2^3$. That single equivalence is the whole foundation of the topic, and the fastest way to answer almost any logarithm question is to translate it straight back into a statement about indices. The base must be positive and not $1$ (a base of $1$ gives only $1^y = 1$, so it could never equal any other number), and the number $x$ inside the log must be positive, because a positive base raised to any real power is always positive, so there is no power of $a$ that equals a zero or negative number.

Converting between index form and log form

Because the two forms say the same thing, every logarithm statement is an index statement in disguise, and switching between them is the core skill. The pattern is fixed: the base stays the base, and the index and the log are the same number. Going from $3^4 = 81$ to log form, the base $3$ stays put, the index $4$ becomes the value of the log, and $81$ is the number: $\log_3 81 = 4$. Going the other way, $\log_{10} 1000 = 3$ says the base is $10$, the index is $3$ and the number is $1000$, so $10^3 = 1000$.

To evaluate a logarithm with nice numbers, set it equal to $x$, rewrite as an index equation, and match powers of the base, exactly the index-matching method from the previous page. For $\log_2 32$, let $x = \log_2 32$, so $2^x = 32 = 2^5$, giving $x = 5$. The same routine handles negative and fractional answers: $\log_2 \frac{1}{8}$ gives $2^x = \frac{1}{8} = 2^{-3}$, so the log is $-3$; and $\log_9 3$ gives $9^x = 3$, that is $3^{2x} = 3^1$, so $x = \frac{1}{2}$.

The logs of 1 and of the base

Two values come up constantly and are worth knowing on sight, and both follow immediately from the definition:

• $\log_a 1 = 0$, because $a^0 = 1$. The log of $1$ is always $0$, whatever the base.
• $\log_a a = 1$, because $a^1 = a$. The log of the base itself is always $1$.

A third, $\log_a \sqrt{a} = \frac{1}{2}$ (because $\sqrt{a} = a^{1/2}$), shows the same idea with a fractional index. These are not separate facts to memorise so much as the definition applied to the easiest possible numbers, and they are the reason a term like $\log_3 3$ silently collapses to $1$ in the middle of a longer calculation.

The three laws of logarithms

Because logarithms are indices, the three index laws for combining powers become three laws for combining logarithms of the same base:

• Product law: the log of a product is the sum of the logs, $\log_a (xy) = \log_a x + \log_a y$.
• Quotient law: the log of a quotient is the difference of the logs, $\log_a \dfrac{x}{y} = \log_a x - \log_a y$.
• Power law: the log of a power is the index times the log, $\log_a x^n = n \log_a x$.

The reason they hold is the reason the topic is coherent: a log is an index, and these mirror the index laws exactly. Multiplying powers adds indices, so the log of a product adds the logs; dividing subtracts; raising a power to a power multiplies, so the log of a power multiplies. (Formally, write $x = a^{\log_a x}$ and $y = a^{\log_a y}$; then $xy = a^{\log_a x + \log_a y}$, and reading off the index gives the product law, with the other two following the same way.) Two cautions are worth fixing now, because they are the commonest log errors. The product law is about the log of a product, not a product of logs: $\log_a x + \log_a y = \log_a (xy)$, never $\log_a x \cdot \log_a y$. And the power law only lifts an index off the whole number inside: $\log_a x^2 = 2 \log_a x$, but $\left(\log_a x\right)^2$ is just that log squared and does not simplify.

Expanding and contracting log expressions

The laws run in both directions, and exam questions ask for each. To expand a single logarithm of a complicated expression into separate logs, peel it apart with the laws: split the quotient first, then the product, then drop the powers. Take $\log_a \dfrac{xy^2}{z}$ with a general base $a$:

To contract (or condense) several logs into one, reverse every step: rewrite any coefficient as an index with the power law, then combine sums into a product and differences into a quotient. For example, $2\log_a 3 + \log_a 4 - 3\log_a 2$ becomes $\log_a 3^2 + \log_a 4 - \log_a 2^3 = \log_a \dfrac{9 \times 4}{8} = \log_a \dfrac{9}{2}$. A frequent trap when contracting is sign and coefficient order: a coefficient must become a power before you combine, and only a leading coefficient of $1$ can be combined directly.

Change of base

A calculator's $\log$ button gives only base $10$ (and $\ln$ gives base $e$, the next page's topic). To evaluate a logarithm in any base, the change-of-base formula rewrites it using a base the calculator knows. For bases $a$ and $b$,

Choosing $b = 10$ turns any logarithm into a quotient of base-$10$ logs you can key in directly: $\log_2 5 = \dfrac{\log_{10} 5}{\log_{10} 2} = 2.322$ (to $4$ significant figures). The formula comes straight from the definition: if $y = \log_a x$ then $x = a^y$; taking $\log_b$ of both sides gives $\log_b x = y \log_b a$ by the power law, and dividing by $\log_b a$ isolates $y$. Beyond evaluation, change of base is what finally lets you produce a decimal for the exact answers logarithms give to exponential equations.

Solving log and index equations

Two equation types now sit within reach. A logarithmic equation is solved by converting to index form. If the unknown is the number, $\log_3 x = 4$ becomes $x = 3^4 = 81$; if the unknown is the base, $\log_x 8 = 3$ becomes $x^3 = 8$, so $x = 2$ (and you check the base is positive and not $1$). An index equation where the two sides cannot be matched to one base, such as $a^x = k$, is exactly what logarithms were built for. The index-matching trick of the previous page fails for $5^x = 18$ because $18$ is not a power of $5$, so instead take a logarithm of both sides, or equivalently rewrite directly in log form:

The exact answer is $\log_5 18$; the decimal comes from change of base. This single move, "take logs of both sides", is the master key for every exponential model in the course, from population doubling to radioactive decay.

Why logarithmic scales exist

Logarithms are not just an algebra exercise; they are how science tames quantities that span enormous ranges. Each of these scales is "the index of $10$", so a step of $1$ on the scale means a factor of $10$ in the underlying quantity:

• pH measures acidity as $\text{pH} = -\log_{10}[\mathrm{H}^+]$, where $[\mathrm{H}^+]$ is the hydrogen-ion concentration. Pure water has $[\mathrm{H}^+] = 10^{-7}$, giving $\text{pH} = 7$. A drop of $2$ pH units (from $5$ to $3$) is not a small change: it is a factor of $10^2 = 100$ more acidic.
• Loudness in decibels is $L = 10\log_{10}\dfrac{I}{I_0}$, comparing a sound's intensity $I$ to a reference $I_0$. An intensity ratio of $10^8$ gives $80$ dB. Doubling the intensity adds only $10\log_{10} 2 \approx 3.01$ dB, which is why two identical speakers sound only a little louder than one.
• Earthquakes on the Richter scale use $M = \log_{10}\dfrac{A}{A_0}$, where $A$ is the measured wave amplitude. A magnitude $6$ quake has $10^{6-4} = 100$ times the amplitude of a magnitude $4$ quake, despite the labels looking close together.

The growth and decay pair below makes the logarithm's role visual. The curve $y = \log_2 x$ is the reflection of the exponential $y = 2^x$ in the line $y = x$: the two functions undo each other. The exponential turns an index into a number ($3 \mapsto 2^3 = 8$); the logarithm turns the number back into the index ($8 \mapsto \log_2 8 = 3$). Every point $(p, q)$ on one curve has its mirror $(q, p)$ on the other, which is exactly the index-and-log swap.

This page builds directly on indices and index laws; if "the log is the index" feels slippery, the surest fix is to be fluent with the index laws first, because every log law here is one of them re-read.

How exam questions ask about logarithms

The command words point you to the exact skill:

• "Evaluate" or "find the value of" a logarithm with nice numbers means convert to an index equation and match powers of the base. $\log_2 32 = 5$ because $2^5 = 32$.
• "Write in index form" or "write in logarithmic form" is a direct conversion: keep the base, swap the index and the log. $\log_3 81 = 4 \iff 3^4 = 81$.
• "Expand" (or "write in terms of $\log x$, $\log y$, ...") means apply the laws forwards: split products into sums, quotients into differences, and bring powers to the front.
• "Express as a single logarithm" or "simplify" means apply the laws in reverse: turn coefficients into indices first, then combine into one log.
• "Solve" a log equation means convert to index form; "solve" an index equation that will not match bases means take logs of both sides.
• "Evaluate, correct to ... significant figures" for a non-base-$10$ log signals the change-of-base formula, log of the number over log of the base.
• "Hence" after an exact log answer (like $\log_5 18$) usually means "now use change of base to give a decimal".