If $\log 3 = 0.477$ and $(1000)^x = 3$, then $x$ equals

### Concept & Formula The core concept is to solve for an unknown exponent by applying logarithms to both sides of an exponential equation. This allows us to use the power rule to bring the exponent down as a multiplier. Formula used: Power Rule: $\log(a^b) = b \log a$ Base 10 Identity: $\log_{10} 1000 = \log_{10} (10^3) = 3$ ### Step-by-Step Solution Given the equation: $$ (1000)^x = 3 $$ Step 1: Take the common logarithm (base 10) of both sides. $$ \log((1000)^x) = \log 3 $$ Step 2: Apply the power rule to bring $x$ to the front. $$ x \log 1000 = \log 3 $$ Step 3: Evaluate $\log 1000$. Since $1000 = 10^3$, its base-10 logarithm is $3$. Substitute this and the given value for $\log 3$ ($0.477$). $$ x(3) = 0.477 $$ Step 4: Solve for $x$ by dividing both sides by $3$. $$ x = \frac{0.477}{3} $$ $$ x = 0.159 $$ ### Exam Strategy & Shortcut **Mental Exponent Conversion:** Recognize immediately that $1000$ is $10^3$. Therefore, $(1000)^x$ is simply $10^{3x}$. If $10^{3x} = 3$, converting this to logarithmic form directly yields $3x = \log 3$. You can instantly bypass the intermediate steps, plug in $0.477$, and mentally divide by $3$ to get $0.159$. ### Common Pitfall A common mistake is attempting to divide the logarithm arguments directly or confusing the base. Some students might incorrectly assume $x = \frac{3}{1000}$ or mishandle the base $10$ properties, leading to answers off by decimal places, like option (a) or (b). ### Final Answer **Therefore, the correct answer is 0.159.**