Question 277175
What do the underscores, "_", mean?<br>
And is the equation
{{{log((2x)) + log((4x)) = 6}}}
or
{{{log(2, (x)) + log(4, (x)) = 6}}}?<br>
Please repost your question making it clear. If needed use more English and less algebraic notation. For example the second equation would be:
base 2 log of x + base 4 log of x = 6
*******
Now that I know what the equation is...
{{{log(2, (x)) + log(4, (x)) = 6}}}
To solve equations where the variable is in the argument(s) of logarithms, you usually start by transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)<br>
Your equation, with the "non-logarithmic" term of 6 on the right, makes the second form more difficult to achieve. So we will aim for the first form.<br>
The first form requires that one side is a single logarithm. So somehow we need to combine your two logarithms into one. Your two logarithms are not like terms so they cannot be added. In addition, a property of logarithms, {{{log(a, (p)) + log(a, (q))}}}, can be used to combine two logarithms which have a "+" between them. But this property requires that the bases of the two logarithms be the same. And your bases are different so we cannot use this property (yet).<br>
So we need the bases the same. Fortunately there is a base conversion for logarithms, {{{log(a, (p)) = log(b, (p))/log(b, (a))}}}, which can be used to convert a logarithm of one base, "a", into an expression of another base, "b". We will use this to convert your base 4 logarithm into base 2:
{{{log(2, (x)) + log(2, (x))/log(2, (4)) = 6}}}
And the denominator is a logarithm we can do "by hand". Since {{{2^2 = 4}}}, {{{log(2, (4)) = 2}}}:
{{{log(2, (x)) + log(2, (x))/2 = 6}}}
or
{{{log(2, (x)) + (1/2)log(2, (x)) = 6}}}
These two logarithms are like terms so we can go ahead and add them:
{{{(3/2)log(2, (x)) = 6}}}
The only thing left to do in order to achieve the desired form is the get rid of the 3/2. We can accomplish this by multiplying both sides by 2/3:
{{{log(2, (x)) = 4}}}
We finally have the desired form. With this form the next step to rewrite the equation in exponential form:
{{{x = 2^4}}}
which simplifies to
{{{x = 16}}}
And we have the answer.<br>
Checking the answer is important (not just a good idea) with logarithmic equations. Always check using the original equation:
{{{log(2, (x)) + log(4, (x)) = 6}}}
Checking x = 16:
{{{log(2, (16)) + log(4, (16)) = 6}}}
Since {{{2^4 = 16}}} and {{{4^2 = 16}}}, {{{log(2, (16)) = 4}}} and {{{log(4, (16)) = 2}}}. This gives us:
{{{4 + 2 = 6}}} Check.