Maths with Lemon

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.

\[\boxed{\log_b x=y\iff b^y=x}\]

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

Worked example

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

\[2^3=8\]

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

Special notation: \(\log_{10}x=\log x\) (common logarithm) and \(\log_e x=\ln x\) (natural logarithm).

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 \(r\):

Product rule
\[\log_b(MN)=\log_b M+\log_b N\]
Quotient rule
\[\log_b\!\left(\frac MN\right)=\log_b M-\log_b N\]
\[\log_b(M^r)=r\log_b M\]
Worked example

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

\[3+2\log_2 x-\log_2 y\]

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:

\[y=\log_b x\quad\longleftrightarrow\quad x=b^y\]
Domain
\[x>0\]
Range
\[y\in\mathbb R\]

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 \(x=2\).

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:

\[\log_2(x-1)+\log_2(x+1)=3\]

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

\[\log_2(x^2-1)=3\]

Convert to exponential form and solve:

\[x^2-1=8\quad\Rightarrow\quad x=\pm3\]

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

\[\boxed{x=3}\]
For equations such as \(2^x=7\), take logarithms: \(x=\frac{\ln 7}{\ln 2}\).

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:

\[\log_b x=\frac{\ln x}{\ln b}\]
Worked example: growth

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

\[500(1.08)^t=800\]
\[t=\frac{\ln(1.6)}{\ln(1.08)}\approx6.11\]

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

Revision Checklist

Definitions

Convert between logarithmic and exponential forms.

Algebra

Apply the product, quotient and power laws.

Graphs

State the domain, range and asymptotes.

Equations

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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