Question 270142
{{{log(4, (y)) = log(2, (x)) + log(2, (9)) - log(2, (3))}}}
To express y in terms of x we will need to get y out of the argument of the logarithm. And to get a variable out of the argument of a logarithm you usually start by transforming the equation into one of these forms:
log(variable-expression) = other-expression
or
log(variable-expression) = log(other-expression)
(with the bases of the logarithms the same in the second form).<br>
To reach either of these forms from your equation we will need to have the same bases for each logarithm. So we need to:<ol><li>Make the bases of the logarithms equal.</li><li>Transform the new equation into one of the forms above.</li><li>Solve the equation from step #2 for y.</li></ol>
Let's see this in action.
1. Same bases. Since there is only one logarithm of base 4 we will change it to match the base of the others (base 2). There is a formula for converting bases of logarithms. To change a base "a" logarithm to an expression with base "b" logarithms: {{{log(a, (x)) = log(b, (x))/log(b, (a))}}}. To change {{{log(4, (y))}}} into an equivalent expression of base 2 logarithms: {{{log(4, (y)) = log(2, (y))/log(2, (4))}}}. Substituting this base 2 expression in for the base 4 logarithm our equation becomes:
{{{log(2, (y))/log(2, (4)) = log(2, (x)) + log(2, (9)) - log(2, (3))}}}
Fortunately, since {{{2^2 = 4}}} then {{{log(2, (4)) = 2}}}. So our equation simplifies to:
{{{log(2, (y))/2 = log(2, (x)) + log(2, (9)) - log(2, (3))}}}<br>
2. Transform the equation into one of the forms above. Since every term of our equation is a logarithm, I'll going to work towards the second form. This will require that we combine the three logarithms on the right into a single logarithm. Fortunately we have properties of logarithms which allow us to do just that:<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>
Since there is a "+" between the first two logarithms we'll use the first property to combine them:
{{{log(2, (y))/2 = log(2, (x*9)) - log(2, (3))}}}
or
{{{log(2, (y))/2 = log(2, (9x)) - log(2, (3))}}}
Since there is a "-" between the remaing two logarithms, I'll use the second property to combine them:
{{{log(2, (y))/2 = log(2, (9x/3))}}}
which simplifies to
{{{log(2, (y))/2 = log(2, (3x))}}}
We are close to the second form. But the 2 in the denominator needs to go. If we multiply both sides by 2 we get:
{{{log(2, (y)) = 2*log(2, (3x))}}}
The 2 is still in the way but it is in a better place. Now we can use a third property of logarithms, {{{q*log(a, (p)) = log(a, (p^q))}}}, which allow us to move a coefficient into the argument as an exponent:
{{{log(2, (y)) = log(2, ((3x)^2))}}}
which simplifies to
{{{log(2, (y)) = log(2, (9x^2))}}}
We have finally achieved the second form (log(...) = log(...)).<br>
3. Solve the equation for y. This requires that we get y out of the argument on the left. With the second form this is pretty easy. The equation says that the base 2 log of y is equal to the base 2 log of {{{9x^2}}}. If their logs are equal then they are equal. In other words:
{{{y = 9x^2}}}
which expresses y in terms of x.