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Logarithms

Applications

A logarithmic function can be used to transform an exponential function into a linear one. For example, by applying ln to both sides of y equals 5 times 10 to the x we get

the natural log of y equals the natural log of 5 plus the natural log of 10 to the power x which equals x times the natural log of 10 plus the natural log of 5. .

Example 3.

Find x such that     e raised to the power 1 plus 2x equals 20.

Click on the question mark to see the next step in the solution process:

Hint: Apply ln to both sides

Then the natural log of e to the (1+2x) equals the natural log of 20.

The natural log and e are inverse functions, so, 1+2x = the natural log of 20.

Subtracting 1 from both sides we get 2x = the natural log of 20 - 1.

Dividing both sides by 2 gives x = frac{ln20-1}/{2}.

Solving we find x = 1.00 to 2 decimal places.

More Examples

Exercise 3.

Solve the equation and round your answer to two decimal places:



Equation
Answer
exp( + x) =
x = (2 d.p.)

Example 4.

Solve the equation    the natural log of (3x+1) equals 2.

Click on the question mark to see the next step in the solution process:

Hint: Apply exp to both sides

This gives e raised to the natural log of (3x+1) equals e squared.

e and ln are inverses so cancelling these we obtain 3x+1 = e squared.

Subtracting one from both sides gives 3x = esquared minus 1.

x = frac{e^2-1}/{3}.

Solving we find x = 2.13 to 2 decimal places.



More Examples

Exercise 4.

Solve the equation and round your answer to two decimal places:



Equation
Answer
ln( x + ) =
x =

Example 5.

Solve the equation    10^(x^2) = 20.

Click on the question mark to see the next step in the solution process:

Hint: Apply ln to both sides

This gives ln(10^(x^2)) = 20.

Bringing down the power of the logarithm, x^2 * ln(10) = 20.

Dividing by ln 10, x^2 = frac{ln(20)}/{ln{10)}.

Take the square root of both sides, x = plus or minus the square root of frac{ln(20)/ln(10)}.

x = plus or minus 1.14 to 2 decimal places.

More Examples

Exercise 5.

Solve the equation and round your answer to two decimal places:



Equation
Answer

     x
     =

x = (2 d.p.)

 

All logarithmic functions can be transformed to a function involving the natural log.


Example 6.

Suppose    y equals log x base 10. .

Then    10 to the y equals x.

By taking ln of both sides we can solve for y.

Click on the question marks to see the step-by-step solution:

y*ln10=lnx.

y=frac{ln(x)}/{ln(10)}.

More Examples

Exercise 6.

Express the function in terms of the natural log, using fractional answers, not decimals:

e.g. The function  y =(1/2)log2x  expressed in terms of the natural log is y =(1/2)lnx/ln2

Don't type any blank spaces in your answer, and put brackets around any fractional numbers.



Function
Answer
y = log

x
  y =

 

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