Question Video: Solving Exponential Equations Mathematics

Video Transcript

Solve three to the power of 𝑥 equals 11 for 𝑥, giving your answer to three decimal places.

There’re two ways that we can solve this equation using logarithms. Firstly, we can use the fact that if 𝑎 to the power of 𝑥 is equal to 𝑏, then 𝑥 is equal to log base 𝑎 of 𝑏. In this question, the constants 𝑎 and 𝑏 are three and 11, respectively. This means that 𝑥 is equal to log base three of 11. We can type the right-hand side directly into our scientific calculator, giving us 2.182658 and so on. As we want the answer given to three decimal places, the deciding number is the six. When the deciding number is five or greater, we round up. Therefore, 𝑥 is equal to 2.183. We can check this answer by substituting our value back into the original equation. Three to the power of 𝑥 is equal to 11.

An alternative method to solve this question would be to take logs of both sides first. We recall that a logarithm written without a base is log base 10. One of our laws of logarithms states that log 𝑥 to the power of 𝑛 is equal to 𝑛 multiplied by log 𝑥. As the exponent on the left-hand side of our equation is 𝑥, this can be rewritten as 𝑥 multiplied by log three. This is equal to log 11. We can then divide both sides of our equation by log three such that 𝑥 is equal to log 11 divided by log three. Once again, we get an answer rounded to three decimal places of 2.183.