# Logarithmic Functions | Maths with Lemon

Source: https://mathswithlemon.io/aa-functions-logarithmic.html


### Logarithmic Functions

Learn the meaning of logarithms, their graphs, laws, equations and applications in IB Mathematics: Analysis and Approaches (SL and HL).


#### Understanding Logarithms


##### What you need to know

- Laws of exponents
- Exponential functions
- Solving equations
Review Exponential Functions before starting.


##### Key Points

A logarithm tells us the power to which a base must be raised to obtain a given number.

Here \(b>0\), \(b\ne1\), and \(x>0\).

Write \(\log_2 8=3\) in exponential form:

Similarly, \(\log_3\frac19=-2\), since \(3^{-2}=\frac19\).


#### Laws of Logarithms


##### Skills

- Expand logarithmic expressions
- Combine logarithms
- Use logarithm laws correctly
Your logarithms notes


##### Key Points

For positive \(M,N\), valid base \(b\), and real \(r\):

Expand \(\log_2\left(\frac{8x^2}{y}\right)\) for \(x,y>0\):

Do not write \(\log_b(M+N)=\log_b M+\log_b N\). This is not a logarithm law.


#### Graphs of Logarithmic Functions


##### What you need to know

- Domain and range
- Vertical asymptotes
- Inverse relationship
- Transformations

##### Key Points

The logarithmic function is the inverse of an exponential function:

The graph passes through \((1,0)\), has vertical asymptote \(x=0\), and increases when \(b>1\) (decreases when \(0 Transformation example \[f(x)=\ln(x-2)+3\] Shift \(y=\ln x\) right 2 and up 3. The domain is \(x>2\); the vertical asymptote is \(x=2\).

Shift \(y=\ln x\) right 2 and up 3. The domain is \(x>2\); the vertical asymptote is \(x=2\).


#### Solving Logarithmic Equations


##### Approach

- Check the domain
- Combine logarithms if needed
- Convert to exponential form
- Check solutions in the original equation
Your video: solving logarithmic equations


##### Worked example

Solve:

Both arguments must be positive, so \(x>1\). Combine the logarithms:

Convert to exponential form and solve:

Only \(x=3\) satisfies the original domain.


#### Change of Base and Applications


##### What you need to know

- Change-of-base formula
- Solving exponential models
- Interpreting answers in context

##### Key Points

Use the change-of-base formula to evaluate logarithms with any valid base:

A population grows according to \(P(t)=500(1.08)^t\). Find when it first reaches 800.

The model reaches 800 after approximately 6.11 years .

IB exam tip: distinguish the exact logarithmic solution from the rounded contextual answer.


#### Extra


##### Material and references

- Hodder Mathematics AA SL ISBN: 9781510462359 Chapters 1A, 1B, 12A, 12B (as listed on your existing page)
- Exponential Functions
- Logarithms notes
- Exponential equations presentation

##### Revision Checklist

Convert between logarithmic and exponential forms.

Apply the product, quotient and power laws.

State the domain, range and asymptotes.

Solve and reject invalid solutions.

For HL extension, consider more complex equations and transformations, while keeping the same domain restrictions.


### IB Past Paper Problems

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