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Solving equations when the pronumeral is an exponent

CraigsMaths
Mathematics Teaching for Learning
Engineering Maths B, Exponents, Logarithms
September 26th, 2008 by Craig Rose
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Engineering Maths B, Exponents, Logarithms
September 26th, 2008 by Craig Rose
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If you have an equation and want to solve for an unknown in the exponent simply take the logarithm of both sides and apply the indices/logarithms laws.

An example follows:

Solve for $x$ :

$4^{\left(3-x\right)}=2\sqrt{2}$

Take the a logarithm of both sides. At this stage the base we use does not matter. We will choose a base later that will make calculations easy.

$\log{4^{\left(3-x\right)}}=\log{2\sqrt{2}}$

Apply the $\log{a^{b}}=b\log{a}$ indice law to the left side:

$\left(3-x\right)\log{4}=\log{2\sqrt{2}}$

Now get $x$ on it’s own:

$x=3-\frac{\log{2\sqrt{2}}}{\log{4}}$

Time to choose a base that will make evaluating the logarithms easy. We must choose the same base for each logarithm. In this case we choose a base of $2$ .