Logarithm and Exponential Question: A Maths Question

Question:

Solve (4x)^{\lg 5} = (5x)^{\lg 7}

Solution:

4^{\lg 5}\cdot x^{\lg 5}=5^{\lg 7}\cdot x^{\lg 7}

\displaystyle\frac{4^{\lg 5}}{5^{\lg 7}}=\frac{x^{\lg 7}}{x^{\lg 5}}=x^{\lg 7-\lg 5}

Using calculator, and leaving answers to at least 4 s.f.,

0.6763=x^{0.1461}

Lg both sides,

\lg 0.6763=0.1461\lg x

\lg x=\frac{\lg 0.6763}{0.1461}=-1.1626

x=10^{-1.1626}=0.0688 (3 s.f.)

Check answer (to prevent careless mistakes):

LHS=(4\times 0.0688)^{\lg 5}=0.406

RHS=(5\times 0.0688)^{\lg 7}=0.406

Since LHS=RHS, we have checked that our answer is valid.

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