QUESTION

# Write $4\log 4+2\log 5-\log 15$ as a single logarithm.

Hint: We will use some basic forms of logarithm like to write the given logarithm into a single term logarithm as $a\log b$ can be written as $\log ({{b}^{a}})$, and $\log a+\log b$ can be written as $\log (ab)$, and $\log a-\log b$ can be written as $\log \dfrac{a}{b}$.
We have given the logarithm equation as $4\log 4+2\log 5-\log 15$ and we have to write it in a single logarithm form. We know that $a\log b$ can be written as $\log ({{b}^{a}})$, and $\log a+\log b$ can be written as $\log (ab)$, and $\log a-\log b$ can be written as $\log \dfrac{a}{b}$. We will use these three basic formulas of logarithm to write the given expression of logarithm into a single logarithm. So, we can write $4\log 4$ as $\log ({{4}^{4}})$ and we can write$2\log 5$ as $\log ({{5}^{2}})$. Therefore $\log ({{4}^{4}})=log256$ and $\log ({{5}^{2}})=log25$.
So, we get logarithm expression as = $\log 256+\log 25-\log 15$, Therefore applying the rule $\log a+\log b$ can be written as $\log (ab)$, we get the modified equation as = $log\text{ }\left( 256x25 \right)\text{ }-\text{ }log\text{ }15$, now we apply the rule $\log a-\log b$ can be written as $\log \dfrac{a}{b}$ to get modified equation as = $\log \left( \dfrac{\left( 256\times 25 \right)}{15} \right)$ = $\log \left( \dfrac{6400}{15} \right)$ = $\log \left( \dfrac{1280}{3} \right)$
Therefore, given the expression $4\log 4+2\log 5-\log 15$ can be written as $\log \left( \dfrac{1280}{3} \right)$.
Note: Using direct logarithmic formula to reduce any logarithmic function will reduce our effort. Logarithmic functions are very helpful for calculating tedious math equations that we cannot calculate with normal mathematical steps. Only care one has to take is of the logarithmic bases. It can be exponential or base 10. Antilogarithm is also a concept that goes hand in hand with logarithm. If we do not use direct formula of logarithm like $a\log b$ can be written as $\log ({{b}^{a}})$, $\log a+\log b$ can be written as $\log (ab)$, then the chances of error in our solution will be increased and also, we have to add unnecessary lines which will consume our time.