Example 7 from Algebra and Trigonometry, 6.3 Logarithmic Functions

Rewriting and Solving a Real-World Exponential Model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation $10^{x} = 500$ represents this situation, where $x$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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Solution
We begin by rewriting the exponential equation in logarithmic form.

$\begin{array}{l} 10^{x} & = 500 \\ \log (500) & = x & Use the definition of the common log . \end{array}$
Next we evaluate the logarithm using a calculator:
- Press [LOG].
- Enter $500,$ followed by [ ) ].
- Press [ENTER].
- To the nearest thousandth, $\log (500) \approx 2.699.$
The difference in magnitudes was about $2.699.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

The amount of energy released from one earthquake was $8,500$ times greater than the amount of energy released from another. The equation $10^{x} = 8500$ represents this situation, where $x$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solution

Your turn

Solution

Your turn