Question

how are the equations 2^7 = 218 and log₂(218) = 7 related?

• to answer the question above, yulissa says:

oi know that 2^x and log₂x are inverses, so all the numbers in the equations are the same.

• what do you think yulissa means?

Explanation:

Step1: Recall inverse - function property

The exponential function \(y = a^{x}\) and the logarithmic function \(y=\log_{a}x\) are inverse functions. If \(y = a^{x}\), then \(x=\log_{a}y\).
For the equation \(2^{7}=218\), the base is \(a = 2\), the exponent is \(x = 7\), and the result is \(y = 218\).
For the equation \(\log_{2}(218)=7\), the base is \(a = 2\), the argument of the logarithm is \(y = 218\), and the result of the logarithm is \(x = 7\).

Step2: Analyze Yulissa's statement

Yulissa means that because \(y = 2^{x}\) and \(y=\log_{2}x\) are inverse functions, the numbers in the two equations are just in different positions. In the exponential equation \(2^{7}=218\), the base \(2\), the exponent \(7\), and the result \(218\) are related. In the logarithmic equation \(\log_{2}(218)=7\), the base \(2\) is the same as in the exponential equation, the number \(218\) which was the result in the exponential equation is now the argument of the logarithm, and the exponent \(7\) in the exponential equation is the result of the logarithmic equation.

Answer:

Yulissa means that since \(y = 2^{x}\) and \(y=\log_{2}x\) are inverse functions, the base, the exponent, and the result in the exponential equation \(2^{7}=218\) are re - arranged in the logarithmic equation \(\log_{2}(218)=7\) such that the base remains the same, the result of the exponential equation becomes the argument of the logarithm, and the exponent of the exponential equation becomes the result of the logarithm.