Define logarithms as the inverse of exponentials, apply the laws of logarithms, sketch logarithmic graphs and solve exponential equations using logs

What this dot point is asking

VCAA wants you to use logarithms as the inverse operation to exponentials, apply the three core log laws, and solve exponential equations.

Definition

$\log_b x = y \iff b^y = x$.

Equivalently $b^{\log_b x} = x$ and $\log_b(b^y) = y$. Logarithms and exponentials are inverse functions.

Two bases dominate:

• Common log ($\log_{10}$, written $\log$): scientific notation, decibels, pH.
• Natural log ($\log_e$, written $\ln$): calculus, continuous growth.

The laws of logarithms

For any positive base $b \neq 1$ and positive $x, y$:

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Special values: $\log_b 1 = 0$, $\log_b b = 1$, $\log_b(b^x) = x$.

Change of base: $\log_b a = \dfrac{\log_{10} a}{\log_{10} b} = \dfrac{\ln a}{\ln b}$.

Graph of IMATH_17

For $b > 1$:

• Domain: $x > 0$. Range: all real $y$.
• Vertical asymptote: $x = 0$.
• Increasing, concave down.
• x-intercept: $(1, 0)$.

Reflects $y = b^x$ in the line $y = x$.

Solving exponential equations

If $b^x = m$ where $m$ is not a power of $b$, take log of both sides:

Worked example

Simplify $\log_2 24 - \log_2 3 + \log_2 4$.

Apply quotient then product law:

$\log_2 (24/3) + \log_2 4 = \log_2 8 + \log_2 4 = \log_2 32 = 5$.

Common traps

Treating $\log(x + y)$ as $\log x + \log y$. Not a log law.

Forgetting the base. $\log$ alone usually means $\log_{10}$; $\ln$ means $\log_e$.

Taking log of negative numbers. Only defined for positive arguments.

In one sentence

Logarithms invert exponentials ($\log_b x = y \iff b^y = x$); the three laws ($\log_b(xy) = \log_b x + \log_b y$, $\log_b(x/y) = \log_b x - \log_b y$, $\log_b(x^n) = n \log_b x$) plus change of base solve exponential equations whose two sides cannot be reduced to a single base.