Question Video: Solving Exponential Equations Mathematics • Second Year of Secondary School

Video Transcript

Solve two multiplied by three to the power of 𝑥 is equal to five multiplied by four to the power of 𝑥 for 𝑥, giving your answer to three decimal places.

In order to calculate an exponent or power in an exponential equation, we need to use logarithms. There are a couple of ways of approaching this problem. One way would be to get all the terms within exponents of 𝑥 on one side of the equation. We can do this by dividing both sides of the equation by two multiplied by four to the power of 𝑥. On the left-hand side, the twos cancel leaving us with three to the power of 𝑥 over four to the power of 𝑥. On the right-hand side, we’re left with five over two or five-halves.

We recall that the quotient of two constants raised to the same exponent can be rewritten as shown. 𝑎 to the power of 𝑥 divided by 𝑏 to the power of 𝑥 is equal to 𝑎 over 𝑏 all to the power of 𝑥. This means we can rewrite the left-hand side of our equation as three-quarters to the power of 𝑥. We then recall the link between exponentials and logarithms. If 𝑎 to the power of 𝑥 is equal to 𝑏, then 𝑥 is equal to log to the base 𝑎 of 𝑏. As 𝑎 is equal to three-quarters and 𝑏 is equal to five over two, then 𝑥 is equal to log to the base three-quarters of five over two.

Typing this into the calculator gives us negative 3.185081 and so on. As we want our answer to three decimal places, the deciding number is the zero. As this is less than five, we round down. Our value of 𝑥 correct to three decimal places is negative 3.185.

We could’ve used an alternative method from the line three-quarters to the power of 𝑥 is equal to five over two. We could take a log of both sides of the equation, such that log of three-quarters to the power of 𝑥 is equal to log of five over two.

One of our laws of logarithms states that log 𝑎 to the power of 𝑥 is equal to 𝑥 log 𝑎. The left-hand side can be rewritten as 𝑥 multiplied by log of three-quarters. We can then divide both sides of the equation by log of three-quarters. 𝑥 is equal to log of five over two divided by log of three-quarters. Once again, this gives us an answer of negative 3.185081 which we know rounds to negative 3.185. We could check this answer by substituting our value back into both sides of the original equation.