Video: Finding the Solution Set of an Exponential Equation

Find the value of 𝑥 for which 8^(3 − 2𝑥) = 12.7. Give your answer to the nearest tenth.

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Video Transcript

Find the value of 𝑥 for which eight to the power of three minus two 𝑥 is equal to 12.7. Give your answer to the nearest tenth.

In order to calculate the exponent 𝑏 in any equation 𝑎 to the power of 𝑏 equals 𝑐, we need to know the link between exponents and logarithms. If 𝑎 to the power of 𝑏 is equal to 𝑐, then 𝑏 is equal to log 𝑐 to the base 𝑎. In this question, the value of 𝑎 is eight, 𝑏 is equal to three minus two 𝑥, and 𝑐 is equal to 12.7. This means that three minus two 𝑥 is equal to log 12.7 to the base eight.

We could type the right-hand side directly into the calculator giving us 1.222252 and so on. However, it is often easier to rearrange the equation to make 𝑥 the subject first. Subtracting three from both sides of the equation gives us negative two 𝑥 is equal to log 12.7 to the base eight minus three. We can then divide both sides of this equation by negative two. 𝑥 is equal to log of 12.7 to the base eight minus three divided by negative two.

Typing this into our calculator gives us 0.888873 and so on. We are asked to round this number to the nearest tenth, which is the same as to one decimal place. The deciding number is the eight in the hundredths column. If this is five or greater, we round up. The value of 𝑥 to the nearest tenth that satisfies the equation is 0.9. We could check our answer by substituting this value back into the original equation.

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