Question

If $\log x-5\log 3=-2$, then $x$ equals 

• A. $1.25$
• B. $0.81$
• C. $2.43$
• D. $0.8$

Hint:

When you see $a\log b$ inside an equation, convert it to $\log(b^a)$ and combine logs.

Answer 1

The Correct Option is C

(Logs are base 10.) Move the term to the RHS: \[ \log x=-2+5\log 3=\log\!\big(10^{-2}\big)+\log\!\big(3^5\big)=\log\!\Big(\frac{3^5}{100}\Big). \] Hence \(x=\dfrac{3^5}{100}=\dfrac{243}{100}=2.43.\) 

Answer 2

Rather than deriving \( x \) algebraically, substitute each option back into \( \log x-5\log3=-2 \) (base-\( 10 \) logs) and check which one satisfies the equation.

1. Option A (\( 1.25 \)): \( \log1.25\approx0.0969 \), and \( 5\log3\approx5\times0.4771=2.3855 \). Then \( 0.0969-2.3855\approx-2.2886\neq-2 \); rejected.
2. Option B (\( 0.81 \)): \( \log0.81\approx-0.0915 \), so \( -0.0915-2.3855\approx-2.477\neq-2 \); rejected.
3. Option C (\( 2.43 \)): \( \log2.43\approx0.3856 \), so \( 0.3856-2.3855\approx-2.0000 \). This satisfies the equation exactly, since \( 2.43=3^5/100 \) makes \( \log2.43=5\log3-2 \) precisely.
4. Option D (\( 0.8 \)): \( \log0.8\approx-0.0969 \), so \( -0.0969-2.3855\approx-2.4824\neq-2 \); rejected.

Only \( x=2.43 \) satisfies the original equation when substituted back and checked numerically.

Hence, the correct answer is 2.43.