Sure, Erika! To solve the exponential equation [tex]\(50^x = 17\)[/tex], you can follow these steps:
1. Understand the Goal:
The goal is to isolate the variable [tex]\(x\)[/tex] in the equation [tex]\(50^x = 17\)[/tex].
2. Use Logarithms:
To isolate [tex]\(x\)[/tex], we can use logarithms, which will help us deal with the exponent. Taking the natural logarithm (denoted as [tex]\(\ln\)[/tex]) is a common approach. You could also use the logarithm with any other base, such as base 10 (denoted as [tex]\(\log\)[/tex]).
3. Apply the Natural Logarithm to Both Sides:
Take the natural logarithm of both sides of the equation to get:
[tex]\[
\ln(50^x) = \ln(17)
\][/tex]
4. Use the Power Rule of Logarithms:
One of the logarithm properties, known as the power rule, allows us to bring the exponent [tex]\(x\)[/tex] in front of the logarithm:
[tex]\[
x \cdot \ln(50) = \ln(17)
\][/tex]
5. Isolate [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], divide both sides of the equation by [tex]\(\ln(50)\)[/tex]:
[tex]\[
x = \frac{\ln(17)}{\ln(50)}
\][/tex]
6. Calculate the Values:
Now, you can calculate the natural logarithms using a calculator:
[tex]\[
\ln(17) \approx 2.833213344
\][/tex]
[tex]\[
\ln(50) \approx 3.912023005
\][/tex]
So, you get:
[tex]\[
x = \frac{2.833213344}{3.912023005} \approx 0.7242322808748767
\][/tex]
Therefore, the solution to the equation [tex]\(50^x = 17\)[/tex] is [tex]\(x \approx 0.724\)[/tex]. This means you need to raise 50 to approximately 0.724 to get close to 17.
