Solving Logarithms for Geosciences

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Question 1

Solve these equations for x, give your answer as integer or fraction.

a)

$\log_\var{a2}\var{a3}+\log_\var{a2}x = \var{a4}$

$x=$ Expected answer:

First you need to use the law of logarithm to combine the two logs on the left handside to one:

$\log_\var{a2}(\var{a3}\times\var{a2}x) = \var{a4}$

Then use the definition of log to change it to exponient form.

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b)

$\log_\var{b1}(x+\var{b4})-\log_\var{b1}\var{b2} = \var{b3}$

$x=$ Expected answer:

Similar to the first question, you need to use the law of logarithm to combine the two logs on the left handside to one:

Then use the definition of log to change it to exponient form.

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c)

$\log_\var{c1}\var{c2}x-\log_\var{c1}(\var{c3}x-\var{c4}) = \var{c5}$

$x=$ Write your answer as a fraction.Expected answer:

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Advice

We want to solve for x. After simplifying, in each case we end up with $\log_a{f(x)} = b$ , so we raise both sides as a power of $a$ to get $a^{\log_a{f(x)}} = a^b$ which simplifies (by laws of logarithms) to $f(x)=a^b$ . We then solve for x accordingly.

Use of the laws of logarithms is crucial here:

$\log{a} + \log{b} = \log{ab}$

$\log{a} - \log{b} = \log{\frac{a}{b}}$

$\log{a^n} = n\log{a}$

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Solving Logarithms for Geosciences

a)

b)

c)

Advice