Question Video: Solving Exponential Equations Mathematics

Solve 20 = 50(1.029)^(𝑥) for 𝑥, giving your answer to three decimal places.

02:47

Video Transcript

Solve 20 is equal to 50 multiplied by 1.029 to the power of 𝑥 for 𝑥, giving your answer to three decimal places.

In order to calculate the exponent of any exponential function, we will need to use logarithms. Before doing this in this question, we will divide both sides of the equation by 50. The left-hand side simplifies to two-fifths. As a decimal, this is equal to 0.4. On the right-hand side, the 50s cancel. So, we are left with 1.029 to the power of 𝑥.

There are two ways of approaching the problem from this point. Firstly, we could recall that if 𝑎 to the power of 𝑥 is equal to 𝑏, then 𝑥 is equal to log base 𝑎 of 𝑏. In our question, 𝑎 is equal to 1.029 and 𝑏 is equal to 0.4. This means that 𝑥 is equal to log to the base 1.029 of 0.4. Typing this into the calculator gives us negative 32.052194 and so on. As we need to give our answer to three decimal places, the one is the deciding number. As this is less than five, 𝑥 is equal to negative 32.052 to three decimal places. We could check this answer by substituting our value of 𝑥 into the original equation.

An alternative method from the line 0.4 is equal to 1.029 to the power of 𝑥 would be to take logs of both sides. We could then use one of our laws of logarithms to simplify the right-hand side. log 𝑎 to the power of 𝑥 is equal to 𝑥 multiplied by log 𝑎. The right-hand side simplifies to 𝑥 multiplied by log of 1.029. Dividing both sides of the equation by this gives us 𝑥 is equal to log 0.4 divided by log of 1.029. Once again, this gives us negative 32.052 to three decimal places.

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