Question

3 log10 8 – ½ log10 36 = log10 x

Answer 1

Step 1 is to simplify the left side using log rules:

`\begin{aligned}3\log_(10)8 - (1)/(2)\log_(10)36 &= \log_(10) x\\[0.75em]\log_(10)(8^3) - \log_(10)(36^(1/2)) &= \log_(10) x\\[0.75em]\log_(10)(512) - \log_(10)(6) &= \log_(10) x\\[0.75em]\log_(10)\left((512)/(6)\right) &= \log_(10) x\\[0.75em]\log_(10)\left((256)/(3)\right) &= \log_(10) x\end{aligned}`

First we bring the coefficients in front of the logs inside the logs as exponents.

Then we evaluate those powers inside the logs.

Next we use the rule that says the difference to two logs of the same base can be combined into one log, with the insides of the logs being divided.

Then we reduce the fraction in the log.

At this point, since it's the same log base on both sides, we can drop those logs and we know x = 256/3.