```Question 127199
This is an "interesting" problem. There may be another way to solve this, but I used logarithms.
If you are not familiar with logarithms, then you will need to repost your problem because once
I respond it drops off the "easy to find" list that tutors use to answer problems.
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I first wrote this problem as:
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{{{(4^(x^2-x))*(128^x) = 8^3/(1/8)}}}
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On the right side, whenever you divide by a fraction you can invert the fraction and use
the inverted form to multiply the numerator. So dividing the numerator by 0ne-eighth is
the same as multiplying the numerator by 8. This means that {{{8^3/(1/8) = 8^3*8 = 8^4 }}} and
{{{8^4 = 8*8*8*8 = 4096}}}. Substituting 4096 for the right side changes the equation to:
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{{{(4^(x^2-x))*(128^x) = 4096)}}}
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Take the log of both sides:
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{{{log((4^(x^2-x)*128^x)) = log(4096)}}}
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On the left side you can apply the rule that the log of a product is equal to the sum
of the logs of the two factors in the product. Applying this rule leads to:
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{{{log((4^(x^2 - x)))+ log((128^x)) = log 4096}}}
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Next, applying the exponential rule of logarithms, in each of the logs on the left side
you can bring out the exponent as a multiplier of the log to get:
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{{{(x^2-x)*log(4) + x*log 128 = log 4096}}}
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Now recognize that log(4), log(128) and log(4096) are just numbers. You can get them from a
scientific calculator. If you take those logs (I used base 10) you will find that they are:
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log(4) = 0.602059991
log(128) = 2.10720997
log(4096) = 3.612359948
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Substitute these values into the equation in place of log(4), log(128), and log(4096) and
the equation becomes:
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{{{(x^2-x)*(0.602059991) + x*(2.10720997) = 3.612359948}}}
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On the left side do the distributed multiplication and you have:
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{{{x^2*(0.602059991) - x*(0.602059991) + x*(2.10720997) = 3.612359948}}}
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Just to be a little more conventional, let's put the constants in front of the variables in
each of the terms on the left side ... just a slight re-arrangement ...
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{{{(0.602059991)*x^2 - (0.602059991)*x + (2.10720997)*x = 3.612359948}}}
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Notice that we have two terms involving just x as the variable. They can be combine and
when you combine their multipliers ... [-0.602059991 + 2.10720997] ... you reduce the
equation to:
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{{{(0.602059991)*x^2 + (1.505149978)*x = 3.612359948}}}
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Then let's get this into conventional quadratic form by putting all the terms on the left side
and having a zero on the right side. Do this by subtracting 3.612359948 from both sides to get:
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{{{(0.602059991)*x^2 + (1.505149978)*x - 3.612359948 = 0}}}
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And just to make this a little more easy to recognize, let's divide both sides (all the terms)
by the multiplier of the x-squared term. In other words, divide both sides by 0.602059991 and
the equation becomes:
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{{{x^2 + 2.5x - 6 = 0}}}
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That sure makes things easier to see ... and easier to work with.  Now you have this in a
simpler to work with quadratic equation. In fact, with a little thought, this equation can
be either factored or you can use the quadratic formula to solve for x. The left side factors
into:
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{{{(x - 1.5)(x + 4)}}}
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and this makes the equation become:
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{{{(x - 1.5)(x + 4)=0}}}
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This equation will be true if either of the factors on the left side equals zero because a
multiplication by zero on the left side makes the entire left side equal to zero and therefore
equal to the right side. So, one at a time set the two factors equal to zero and you have:
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{{{(x - 1.5) = 0}}}
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add 1.5 to both sides and this becomes:
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{{{x = 1.5}}}
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Then set the other factor equal to zero:
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{{{x + 4 = 0}}}
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and subtract 4 from both sides to get:
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{{{x = -4}}}
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Check both answers by returning to the original equation:
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{{{(4^(x^2-x))*(128^x) = 8^3/(1/8)}}}
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and first replacing x by 1.5 and then by -4 to make sure the equation still balances.
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When x is replaced by 1.5 you have:
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{{{(4^(2.25-1.5))*(128^1.5) = 8^3/(1/8)}}}
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Don't forget that the right side is 4096 so the equation becomes:
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{{{(4^(2.25-1.5))*(128^1.5) = 4096)}}}
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The exponent of the first factor is (2.25 - 1.5 = 0.75) and this makes the equation become:
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{{{4^0.75 * 128^1.5 = 4096}}}
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Calculator time ... raise the two factors on the right side to their respective exponents and
you get:
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{{{(2.828427125)*(1448.154688) = 4096}}}
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and when you do the multiplication on the left side you get 4096, so when you use 1.5 for x
it works.
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Next check when you use -4 for x. The original problem becomes:
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{{{(4^((-4)^2-(-4)))*(128^(-4)) = 8^3/(1/8) = 4096}}}
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The exponent of the first factor on the left side simplifies to give:
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{{{(4^(16+4))*(128^(-4)) = 4096}}}
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Add the 16+4 to get an exponent of 20 and note that the negative exponent of the second factor
makes the second factor equivalent to {{{1/128^4}}}
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this makes the equation become:
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{{{(4^20)*(1/128^4) = 4096}}}
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But 4 to the 20th power is 1.099511628 *10^12 and when this is divided by 128 to the 4th
power (which is 26835456) the result is again 4096. So the original equation reduces to:
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4096 = 4096
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and therefore the original equation holds true when x = -4
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The two answers check. Those answers are x = 1.5 and x = -4
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There probably is an easier way to do this, but this way works.
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Hope this helps you to see one way to do the problem.
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