Question

question 3 (essay worth 10 points) (mc) mike is working on solving the exponential equation 37^x = 12; however, he is not quite sure where to start. solve the equation and use complete sentences to describe the steps to solve. hint: use the change of base formula: \log_b y = \frac{\log y}{\log b}.

Explanation:

Step1: Take log on both sides

To solve the exponential equation \( 37^x = 12 \), we start by taking the logarithm of both sides. Using the common logarithm (base 10) or natural logarithm (base \( e \)) is fine. Let's take the common logarithm (log base 10) of both sides:
\( \log(37^x) = \log(12) \)

Step2: Apply logarithm power rule

The power rule of logarithms states that \( \log(a^b)=b\log(a) \). Applying this rule to the left - hand side, we get:
\( x\log(37)=\log(12) \)

Step3: Solve for x

To isolate \( x \), we divide both sides of the equation by \( \log(37) \):
\( x = \frac{\log(12)}{\log(37)} \)

Step4: Calculate the value (optional, for numerical answer)

Using a calculator, we know that \( \log(12)\approx1.0792 \) and \( \log(37)\approx1.5682 \). Then \( x=\frac{1.0792}{1.5682}\approx0.688 \) (or we can also use the change - of - base formula \( \log_{a}b=\frac{\log b}{\log a} \), here we can think of \( x = \log_{37}(12)=\frac{\log 12}{\log 37} \) which is the same as above).

Answer:

To solve the exponential equation \( 37^x = 12 \):

1. First, take the logarithm of both sides. We can use the common logarithm (base 10) or the natural logarithm (base \( e \)). Using the common logarithm, we have \( \log(37^x)=\log(12) \).
2. Then, apply the power rule of logarithms (\( \log(a^b)=b\log(a) \)) to the left - hand side. This gives us \( x\log(37)=\log(12) \).
3. Next, solve for \( x \) by dividing both sides of the equation by \( \log(37) \). So, \( x = \frac{\log(12)}{\log(37)} \).
4. If we want a numerical approximation, we can calculate the values of \( \log(12) \) and \( \log(37) \) using a calculator. \( \log(12)\approx1.0792 \), \( \log(37)\approx1.5682 \), and \( x=\frac{1.0792}{1.5682}\approx0.688 \).

In summary, the solution to the equation \( 37^x = 12 \) is \( x=\frac{\log(12)}{\log(37)}\approx0.688 \).