To solve the equation [tex]\(-5^x = 80\)[/tex] for [tex]\(x\)[/tex], you need to isolate [tex]\(x\)[/tex]. Let me walk you through the steps:
1. Understand the Equation: The original equation is [tex]\(-5^x = 80\)[/tex].
2. Isolate the Term Involving [tex]\(x\)[/tex]: To make the equation easier to solve, we need to isolate the exponential term [tex]\(5^x\)[/tex].
3. Remove Negative Sign: Since the left side of the equation is negative, multiply both sides of the equation by [tex]\(-1\)[/tex]:
[tex]\[
-(-5^x) = -80
\][/tex]
Which simplifies to:
[tex]\[
5^x = -80
\][/tex]
4. Logarithmic Transformation: Apply the natural logarithm (ln) to both sides of the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
\ln(5^x) = \ln(80)
\][/tex]
5. Use Logarithm Properties: Use the property [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[
x \ln(5) = \ln(80)
\][/tex]
6. Solve for [tex]\(x\)[/tex]: Divide both sides by [tex]\(\ln(5)\)[/tex]:
[tex]\[
x = \frac{\ln(80)}{\ln(5)}
\][/tex]
7. Complex Solution: Since [tex]\(5^x\)[/tex] is real and positive, if we introduce the complex logarithm, we will consider both the real and imaginary parts:
[tex]\[
x = \frac{\ln(80) + i\pi}{\ln(5)}
\][/tex]
Taking all these steps into account, the solution to the equation [tex]\(-5^x = 80\)[/tex] is:
[tex]\[
x = \frac{\ln(80) + i\pi}{\ln(5)}
\][/tex]
Therefore, the step-by-step solution gives us the answer:
[tex]\[ x = \frac{\log(80) + i\pi}{\log(5)}. \][/tex]
