Example 10.37: Approximating a Logarithm Using the Change-of-Base Formula

As a data analyst, you are writing a script to evaluate a logarithmic dataset with base ${}5$. Your software library only has built-in functions for common logarithms (base ${}10$) and natural logarithms (base $e$). According to the Change-of-Base Formula, which of the following represents the correct way to rewrite $\log_5 M$ using natural logarithms so it can be evaluated by the script?

A student worker in the financial aid office is using a growth formula to calculate how long it will take for a student's savings account balance to double. The formula involves the expression $\log_{1.04} 2$. Since their calculator only has a natural logarithm ($\ln$) button, they use the Change-of-Base Formula to rewrite the expression. According to the formula, what value completes the denominator of the following conversion? \log_{1.04} 2 = \frac{\ln 2}{\text{____}}

A laboratory technician is analyzing experimental data recorded using various logarithmic scales. To enter this data into a centralized database that only supports natural logarithms ($\ln$), the technician must apply the Change-of-Base Formula. Match each original recorded expression with its mathematically equivalent natural log conversion.

A data analyst is using a spreadsheet to evaluate the expression $\log_2 100$. Because the spreadsheet software only has a built-in function for natural logarithms ($\ln$), the analyst uses the Change-of-Base Formula to rewrite the expression as $\frac{\ln 2}{\ln 100}$. Is this conversion correct?

A technical documentation specialist is verifying the algebraic logic for a new math library that handles logarithmic conversions. To ensure the documentation accurately reflects the math, the specialist must outline the derivation of the Change-of-Base Formula starting from the expression $y = \log_a M$. Arrange the following algebraic steps in the correct logical sequence to complete this derivation.