```Question 397005
{{{log((16)) +(1/2)log((5))- log((8))}}}
These are not like terms. (Like terms with logarithms have the same base of logarithm and the arguments are the same. All your logairthms are base 10 but the arguments are all different.)<br>
But there are these properties of logarithms<ul><li>{{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}</li><li>{{{log(a, (p)) - log(a, (q)) = log(a, (p/q))}}}</li></ul>which can be used. The properties do not require that the arguments be the same. They do require<ul><li>logarithms of the same base.</li><li>Coefficients of 1 in front of the logarithms.</li></ul>
Except for the middle logarithm, which has a coefficient of 1/2, your logarithms fit the requirements for these properties. And as for the middle logarithm, there is yet another property, {{{q*log(a, (p)) = log(a, (p^q))}}}, which allows us to move a coefficient into the argument as an exponent. So we start by using thie third property on the middle logarithm:
{{{log((16)) + log((5^(1/2)))- log((8))}}}
And since 1/2 as an exponent means square root, we can rewrite this as:
{{{log((16)) + log((sqrt(5)))- log((8))}}}
Now we can go ahead and use the first two properties. First we use the first property on the first two logarithms. (We use the first property because your first two logarithms have a "+" between them.)
{{{log((16*sqrt(5)))- log((8))}}}
Now we use the second property on the remaining two logarithms. (We use the second property now because we have a "-" between the two logarithms.)
{{{log(((16*sqrt(5))/8))}}}
And the fraction reduces:
{{{log((2*sqrt(5)))}}}
This is a single logarithm in simplified form that is equal to your original expression.```