Question 64477
Solve for x:
{{{log3(x-2)+log3(x+4) = 3}}} Applying the product rule for logarithms:{{{log(M)+log(N) = log((MN))}}} we get:
{{{log3((x-2)(x+4)) = 3}}} Simplifying:
{{{log3(x^2+2x-8) = 3}}} Writing this in exponential form:{{{logb(x) = y}}} means {{{b^y = x}}}
{{{3^3 = x^2+2x-8}}} Simplify:
{{{x^2+2x-17 = 0}}} Solving by the quadratic formula:{{{x = (-b+-sqrt(b^2-4ac))/2a}}}
{{{x = (-2+-sqrt(2^2-4(1)(-17)))/2(1)}}} Simplifying:
{{{x = (-2+-sqrt(72))/2}}} Simplifying further:
{{{x = (-2+-sqrt(36*2))/2}}}
{{{x = -1+3sqrt(2)}}} and {{{x = -1-3sqrt(2)}}}
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Solve for x:
{{{e^x + 5 = 8}}} Subtract 5 from both sides.
{{{e^x = 3}}} Take the natural log of both sides.
{{{xln(e) = ln(3)}}} But {{{ln(e) = 1}}}, so:
{{{x = ln(3)}}}
{{{x = 1.0986}}} Approx.