In professional fields such as data science or finance, converting between growth models (exponential) and their scales (logarithmic) is a frequent task. Based on the definition $y = \log_a x$ if $x = a^y$, match each exponential equation with its equivalent logarithmic form.

A laboratory technician is modeling the growth of a bacterial culture that triples every hour. The growth is represented by the exponential equation ${}243 = 3^5$, where 243 is the number of bacteria after 5 hours. Which of the following equations correctly expresses this relationship in logarithmic form?

A financial analyst uses the exponential equation ${}10^2 = 100$ to calculate a growth factor over two periods. True or False: The equivalent logarithmic form of this relationship is ${}2 = \log_{10} 100$.

Logistics Expansion and Logarithmic Form

A biologist tracking a specific cell culture's growth volume notes that it follows the exponential pattern ${}125 = 5^3$. To report this relationship in a research paper using logarithmic form $y = \log_a x$, the biologist must identify which part of the exponential equation becomes the argument of the logarithm. Complete the equivalent logarithmic equation: ${}3 = \log_5$ ____.

A business administrative assistant at a local delivery company is creating an onboarding guide for new route planners. The guide outlines how to convert exponential routing models—such as the base growth equation ${}243 = 3^5$—into logarithmic equations to plot delivery volume on logarithmic scales.

Arrange the steps in the correct chronological order to show how to convert the exponential equation ${}243 = 3^5$ into its equivalent logarithmic form, based on the definition that $y = \log_a x$ if $x = a^y$.

Documenting the Logarithmic Conversion for Environmental Field Reports

Try It 10.36: Converting to Logarithmic Form