Video: Finding the Solution Set of a Logarithmic Equation

Use a calculator to find the value of π‘₯ for which π‘₯^(7.1) = 97.4. Give your answer correct to two decimal places.

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Video Transcript

Use a calculator to find the value of π‘₯ for which π‘₯ to the power of 7.1 is equal to 97.4. Give your answer correct to two decimal places.

There are a number of methods we could use to solve this problem. In this question, we will use logarithms. Our first step is to take logs of both sides of the equation such that log of π‘₯ to the power of 7.1 is equal to log of 97.4. Note that the standard log button on the calculator refers to log to the base 10. We recall that one of our laws of logarithms states that log of π‘₯ to the power of 𝑛 is equal to 𝑛 multiplied by log π‘₯.

We can therefore rewrite the left-hand side as shown, 7.1 multiplied by log π‘₯. Dividing both sides of this equation by 7.1 gives us log π‘₯ is equal to log of 97.4 divided by 7.1. Typing the right-hand side into our calculator gives us 0.280078 and so on. In order to calculate our value of π‘₯, we recall that if log π‘₯ with the base π‘Ž is equal to 𝑏, then π‘₯ is equal to π‘Ž to the power of 𝑏.

As we’re working in base 10, π‘₯ is equal to 10 to the power of 0.280078 and so on. Typing this into the calculator gives us 1.905806 and so on. We’re asked to give our answer to two decimal places. Therefore, the deciding number is the five. When a deciding number is five or greater, we round up. The solution for π‘₯ to the power of 7.1 equals 97.4 is 1.91 to two decimal places. We can substitute this back into our original equation to check our answer.

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