Using logarithms to find x/y when 7²ˣ = 11⁵ʸ

Full worked solution

1. Take logarithms of both sides

   We start by taking the common logarithm (or natural logarithm, either will work) of both sides of the equation, since we can use logarithm power rules to bring down the exponents. Applying the logarithm to both sides:

   $$\log\left(7^{2x}\right) = \log\left(11^{5y}\right)$$

   Note.

   You can use base 10 (log) or base e (ln) for this step — the ratio will be identical regardless of logarithm base.

2. Apply the logarithm power law

   The logarithm power law states $\log (a^b)=b\log (a)$ for any positive $a$. Use this to rewrite both sides by moving the exponents outside the logarithm:

   $$2x \log 7 = 5y \log 11$$

   Note.

   Make sure you apply the power rule correctly to the entire exponent: both the coefficient (2, 5) and the variable (x, y) multiply the log term.

3. Rearrange to isolate the ratio $\frac{x}{y}$

   First, divide both sides by $y$ to combine the $x$ and $y$ terms:

   $$2\left(\frac{x}{y}\right)\log 7 = 5\log 11$$

   Next, divide both sides by $2\log 7$ to solve for $\frac{x}{y}$:

   $$\frac{x}{y} = \frac{5\log 11}{2\log 7}$$

   Note.

   Double-check the order of coefficients: 5 comes from the right-hand side exponent, 2 from the left-hand side exponent.

4. Evaluate the expression to 4 significant figures

   Substitute the numerical values of the logarithms: $\log 11\approx 1.04139$ and $\log 7\approx 0.84510$.

   $$\frac{x}{y} \approx \frac{5 \times 1.04139}{2 \times 0.84510} = \frac{5.20695}{1.69020} \approx 3.081$$

   Therefore, the value of $\frac{x}{y}$ to 4 significant figures is $3.081$.

   Note.

   If you use natural logarithm instead: $\frac{5\ln 11}{2\ln 7}\approx \frac{5\times 2.3979}{2\times 1.9459}=3.081$, which matches the result.

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