Question 813675
{{{log((x)) + log((x^2)) + log((x^3)) + log((x^4)) = 1 + log((0.2)) + log((0.03^2)) + log((0.004^3))}}}
A general procedure for solving these kinds of equations:<ol><li>Use algebra and/or properties of logarithms to transform the equation into one of the following forms:<ul><li>log(expression) = other-expression</li><li>log(expression) = log(other-expression) (Note: The bases of the two logs must match.)</li></ul></li><li>Eliminate the logarithms:<ul><li>If the equation is in the first form, "log(expression) = other-expression", rewrite the equation in exponential form.</li><li>If the equation is in the second form, "log(expression) = log(other-expression)", set the arguments equal.</li></ul></li><li>Now that the logs are gone, solve the equation (using techniques which are appropriate for the type of equation it is).</li><li>Check your solution. This is <b>not</b> optional! A check must be made to see if the bases and arguments of all logs are valid. Any "solution" which make any base or an argument invalid must be rejected! (Note: Valid bases are positive but not 1 and valid arguments are positive.)</li></ol>Let's try this on your equation. First we decide which form we think will be easiest to achieve. With the "non-log" term of 1 (on the right side), it would seem that the second, "all-log" form will be harder to reach. So we will aim for the first form.<br>
Stage 1: Transform
To reach this form, all we need to do is find a way to combine all the logs into a single logarithm. We will start by combining all the logs on each side into single logarithms. For this we will use the {{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}} property:
{{{log((x*x^2*x^3*x^4)) = 1 + log((0.2*0.03^2 * 0.004^3))}}}
Simplifying...
{{{log((x^10)) = 1 + log((0.2*0.0009 * 0.000000064))}}}
{{{log((x^10)) = 1 + log((0.00000001152))}}}
Now we will get the logs on the same side. Subtracting the log on the right we get:
{{{log((x^10)) - log((0.00000001152)) = 1 }}}
And now we can use another property of logarithms, {{{log(a, (p)) - log(a, (q)) = log(a, (p/q))}}}, to combine the remaining logs:
{{{log((x^10/0.00000001152)) = 1 }}}
We have now reached the first form.<br>
Stage 2: Eliminate the logs.
With the first form we just rewrite the equation in exponential form. In general {{{log(a, (p)) = n}}} is equivalent to {{{p = a^n}}}. Using this pattern, and the fact that the base of "log" is 10, we get:
{{{x^10/0.00000001152 = 10^1}}}
which simplifies to:
{{{x^10/0.00000001152 = 10}}}<br>
Srage 3: Solve
Our equation is now an exponential equation. But with only one x term it will not be hard to solve. Multiplying each side by 0.00000001152 we get:
{{{x^10 = 0.0000001152}}}
Now we find the 10th root of each side (remembering both the positive and negative roots):
{{{x = root(10, 0.0000001152)}}} or {{{x = -root(10, 0.0000001152)}}}<br>
Stage 4: Check
Use the original equation to check:
{{{log((x)) + log((x^2)) + log((x^3)) + log((x^4)) = 1 + log((0.2)) + log((0.03^2)) + log((0.004^3))}}}
Checking x = {{{x = root(10, 0.0000001152)}}}:
This can be done by inspection. This value of x is positive. So all the powers of x on the left side will be positive if x is positive. So all the bases arguments are valid. So this value checks out!
Checking {{{x = -root(10, 0.0000001152)}}}:
With x being negative, even powers of x will be positive but odd powers of x will be negative. This makes the 1st and 3rd arguments negative. This is invalid. So we must reject this "solution".<br>
So the only solution to {{{log((x)) + log((x^2)) + log((x^3)) + log((x^4)) = 1 + log((0.2)) + log((0.03^2)) + log((0.004^3))}}} is x = {{{x = root(10, 0.0000001152)}}}.<br>
P.S. Technically, the solution {{{x = root(10, 0.0000001152)}}} should be rationalized. (A decimal is just a disguised fraction and radicals are not supposed to have fractions in them.) So the rest of this is rationalizing the solution:
{{{x = root(10, 0.0000001152)}}}
{{{x = root(10, 1152/10^10)}}}
{{{x = root(10, 1152)/root(10, 10^10)}}}
{{{x = root(10, 1152)/10}}}