DB225: Maths for Diploma Studies: Rearranging Equations – Working with Logarithms Reinforcement Questions

Rearranging Equations – Working with Logarithms Reinforcement Questions

Isolate b in the equation below:

$begin mathsize 14px style fraction numerator L o g left parenthesis b right parenthesis over denominator increment rho subscript omega end fraction space equals space fraction numerator F space minus space empty set C over denominator Q L cubed end fraction end style$

We can start by getting $begin mathsize 14px style L o g left parenthesis b right parenthesis end style$ on its own by moving $begin mathsize 14px style increment rho subscript omega end style$ across the equals sign by multiplying both sides by it:

$begin mathsize 14px style L o g left parenthesis b right parenthesis space equals space fraction numerator F space minus space empty set C over denominator Q L cubed end fraction space increment rho subscript omega end style$

Now we can simply exponentiate both sides to make $fraction numerator size 14px F size 14px space size 14px minus size 14px space size 14px empty set size 14px C over denominator size 14px Q size 14px L to the power of size 14px 3 end fraction size 14px space size 14px increment size 14px rho subscript size 14px omega$ an exponent of 10 and cancel the logarithm of b:

$begin mathsize 14px style b space equals space 10 to the power of open parentheses fraction numerator F space minus space empty set C over denominator Q L cubed end fraction space increment rho subscript omega close parentheses end exponent end style$

Rearrange this equation to make x the subject:

$begin mathsize 14px style phi to the power of negative 1 end exponent space equals space fraction numerator increment b over denominator 10 to the power of open parentheses x close parentheses end exponent end fraction end style$

Our starting point is to move $begin mathsize 14px style 10 to the power of open parentheses x close parentheses end exponent end style$ across the equation by multiplying both sides by it:

$begin mathsize 14px style phi to the power of negative 1 end exponent space 10 to the power of open parentheses x close parentheses end exponent space equals space increment b end style$

Now we can move $size 14px phi to the power of size 14px minus size 14px 1 end exponent$ by dividing both sides by it:

$begin mathsize 14px style 10 to the power of left parenthesis x right parenthesis end exponent space equals space fraction numerator increment b over denominator phi to the power of negative 1 end exponent end fraction end style$

Finally, we take a log10 of both sides to cancel the exponentiation of x and isolate:

$begin mathsize 14px style x space equals space L o g subscript 10 to the power of open parentheses fraction numerator increment b over denominator phi to the power of negative 1 end exponent end fraction close parentheses end exponent end style$

Isolate r in the following equation:

$begin mathsize 14px style l n left parenthesis r right parenthesis space left parenthesis D space plus space C right parenthesis space equals space 2 over 9 space b to the power of x end style$

We start by moving $begin mathsize 14px style left parenthesis D space plus space C right parenthesis end style$ across the equals sign by dividing both sides by it:

$begin mathsize 14px style l n left parenthesis r right parenthesis space equals space fraction numerator begin display style 2 over 9 end style b to the power of x over denominator left parenthesis D space plus space C right parenthesis end fraction end style$

Now we just need to make both sides exponents of e to cancel the natural log of r:

$begin mathsize 14px style r space equals space e to the power of open parentheses fraction numerator begin display style 2 over 9 b to the power of x end style over denominator left parenthesis D space plus space C right parenthesis end fraction close parentheses end exponent end style$

In the equation below, isolate c:

$begin mathsize 14px style L o g left parenthesis c b right parenthesis space equals space increment delta rho end style$

First, we move to cancel the logarithm by making $begin mathsize 14px style increment delta rho space end style$ the exponent of base 10:

$begin mathsize 14px style c b space equals space 10 to the power of open parentheses increment delta rho close parentheses space end exponent end style$

Now we just need to move b across the equals sign to isolate c by dividing both sides by it:

$begin mathsize 14px style c space equals space 10 to the power of left parenthesis increment delta rho right parenthesis space end exponent over b end style$

We continue with equations next. This time looking at how we deal with fractions.

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Maths for Diploma Studies

Rearranging Equations – Working with Logarithms Reinforcement Questions

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