Question

daniel is looking over some data regarding the temperatures of countries and how the temperature relates to the number of mosquitoes. the data are represented by the exponential function f(x) = 2^x, where x is the number of mosquitoes. explain how he can convert this equation to a logarithmic function when the temperature is 64 degrees.

Explanation:

Step1: Recall Exponential to Logarithmic Conversion

The general form of an exponential function is \( y = a^x \), which can be converted to a logarithmic function using the definition of a logarithm: if \( y = a^x \), then \( x=\log_a(y) \). Here, the exponential function is \( f(x) = 2^x \), and we know the temperature (let's assume \( f(x) \) represents the number of mosquitoes related to temperature, and when temperature is 64, we need to find \( x \) or convert the equation). Wait, actually, when the temperature is 64, we can set \( f(x)=64 \) (assuming the output of the function relates to temperature or vice - versa, maybe the function is \( f(x) = 2^x \) where \( f(x) \) is related to temperature? Wait, the problem says "when the temperature is 64 degrees", so we set \( f(x)=64 \) (assuming the function \( f(x) = 2^x \) gives a value related to temperature or the number of mosquitoes, and we need to convert \( 64 = 2^x \) to logarithmic form).

Step2: Apply the Conversion Formula

Given \( 64 = 2^x \), using the logarithm definition: if \( y=a^x \), then \( x = \log_a(y) \). Here, \( a = 2 \), \( y = 64 \), so \( x=\log_2(64) \). We can also calculate \( x \) since \( 2^6=64 \), so \( x = 6 \). But the main task is to convert the exponential equation \( 64 = 2^x \) (derived from \( f(x)=2^x \) and \( f(x) = 64 \)) to a logarithmic function. The general conversion from \( y = 2^x \) to logarithmic form is \( x=\log_2(y) \). When \( y = 64 \), the logarithmic equation is \( x=\log_2(64) \), and solving it gives \( x = 6 \).

Answer:

To convert \( f(x)=2^x \) (with \( f(x) = 64 \) when temperature is 64) to a logarithmic function:

1. Start with the exponential equation: \( 64=2^x \) (since \( f(x) = 64 \) at 64 degrees).
2. Use the logarithm definition (\( y = a^x\Rightarrow x=\log_a(y) \)): For \( a = 2 \), \( y = 64 \), the logarithmic form is \( x=\log_2(64) \).
3. Solve \( \log_2(64) \): Since \( 2^6 = 64 \), \( x = 6 \). The logarithmic function corresponding to \( y = 2^x \) is \( x=\log_2(y) \), and at \( y = 64 \), it is \( \log_2(64)=6 \).