Question

solve the equation. log 4x + log x = 10
x = (round to the nearest whole number as needed.)

Explanation:

Step1: Apply log - product rule

Using the rule $\log a+\log b=\log(ab)$, we can rewrite the left - hand side of the equation $\log 4x+\log x = 10$ as $\log(4x\cdot x)=\log(4x^{2})$. So the equation becomes $\log(4x^{2}) = 10$.

Step2: Rewrite in exponential form

If the base of the logarithm is 10 (common logarithm), then by the definition of logarithms, if $\log y = z$, then $y = 10^{z}$. So, from $\log(4x^{2}) = 10$, we have $4x^{2}=10^{10}$.

Step3: Solve for $x^{2}$

Divide both sides of the equation $4x^{2}=10^{10}$ by 4: $x^{2}=\frac{10^{10}}{4}=\frac{10000000000}{4}=2500000000$.

Step4: Solve for $x$

Take the square root of both sides. Since $x>0$ (because we have $\log x$ in the original equation and the argument of a logarithm must be positive), $x=\sqrt{2500000000}=50000$.

Answer:

$50000$