Evaluate log 1576 divided by log 2.44.

Step-by-step solution

1. 1

   Recognise the change-of-base pattern. For any admissible base $c$,

   $\log_b a=\frac{\log_c a}{\log_c b}$

   A quotient of two logarithms taken in the same base is therefore a single logarithm — and the base $c$ never appears in the answer, so it does not matter whether these are common logs or natural logs.

2. 2

   Read off the base and the argument. Here the numerator's argument is $1576$ and the denominator's is $2.44$, so

   $\frac{\log 1576}{\log 2.44}=\log_{2.44}1576$

   In words: the exponent $x$ for which $2.44^x=1576$.

3. 3

   Evaluate the two logarithms. In base $10$,

   $\log 1576\approx 3.1975562,\qquad \log 2.44\approx 0.3873898$

   Both are safely non-zero, so the quotient is well defined ($2.44\neq 1$).

4. 4

   Divide.

   $\frac{3.1975562}{0.3873898}\approx 8.2541048$

   Carry seven digits through the division: truncating the logs too early is what produces the commonly quoted but wrong value $8.2537$.

5. 5

   Verify by exponentiating. $2.44^{8.2541}\approx 1575.99$, which is $1576$ to within rounding. By contrast $2.44^{8.2537}\approx 1575.43$ — already $0.57$ short — which is how you can tell $8.2537$ is not accurate to four decimals. So the value is $\log_{2.44}1576\approx 8.2541$.