solve exponential equation 10^2x = 7
Check the final answer first, then review the worked steps.
Problem
solve exponential equation 10^2x = 7
Answer
\(x = \frac{\log 7}{2}\)
Step-by-step solution
- Apply the logarithm to both sides: To solve for $x$ in the exponential equation $10^{2x} = 7$, we take the common logarithm (base 10) of both sides: $$\log(10^{2x}) = \log(7)$$
- Use the power rule for logarithms: Apply the property $\log(a^b) = b \cdot \log(a)$ to the left side of the equation: $$2x \cdot \log(10) = \log(7)$$
- Simplify the logarithm: Since $\log(10) = 1$ (because the base of the common logarithm is 10), the equation simplifies to: $$2x \cdot 1 = \log(7)$$ $$2x = \log(7)$$
- Solve for x: Divide both sides by 2 to isolate $x$: $$x = \frac{\log(7)}{2}$$