How do we solve an equation when the unknown is in the exponent?
Exponential equations are solved by taking logarithms of both sides and applying the power law to bring the unknown exponent down.
How to solve exponential equations by taking logs and using the power law, with worked growth-and-decay applications including doubling time and half-life.
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What this dot point is asking
When the unknown sits in the exponent, ordinary algebra cannot isolate it. The logarithm is the tool that "undoes" the exponential, letting you bring the exponent down where you can solve for it.
The core technique
Equations with base
When the base is , taking is especially clean because .
Growth-and-decay applications
Most exam questions dress this technique as a real model - doubling time, half-life, or reaching a target value.
Common errors
Why it matters
Solving exponential equations is the payoff of Topic 4 and a guaranteed exam skill, especially in growth-and-decay contexts. It draws together the log laws, change of base, and the exponential models you differentiated earlier, and it recurs whenever a continuous random variable or model must be inverted for a target value.