Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are used to solve equations involving exponents. Represented as $f(x)={\log }_b(x)$, they determine the power to which a base b must be raised to produce a given number. These functions are widely used in mathematics, science, and engineering to model phenomena such as sound intensity, pH levels, and earthquake magnitudes. Understanding logarithmic functions is essential for simplifying complex calculations and analyzing exponential growth or decay.

1.0Logarithmic Functions Definition

A logarithmic function is the inverse of an exponential function. If:

$y=b^x$

then the logarithmic form is:

$x={\log }_b(y)$

Where:

• b is the base of the logarithm (must be > 0 and ≠ 1),
• y > 0,
• ${\log }_b(y)$ means "the power to which b must be raised to get y."

2.0Logarithmic Functions Formulas

Here are essential logarithmic function formulas:

1. Definition:

${\log }_b(x)=y\iff b^y=x$

2. Natural Logarithm (base e):

$\ln (x)={\log }_e(x)$

3. Common Logarithm (base 10):

$\log (x)={\log }_{10}(x)$

4. Change of Base Formula:

${\log }_b(x)=\frac{{\log }_k(x)}{{\log }_k(b)}(\text{ any base }k)$

3.0Logarithmic Functions Properties

Important logarithmic functions properties:

1. ${\log }_b(1)=0$
2. ${\log }_b(b)=1$
3. ${\log }_b(xy)={\log }_b(x)+{\log }_b(y)$
4. ${\log }_b\left(\frac{x}{y}\right)={\log }_b(x)-{\log }_b(y)$
5. ${\log }_b\left(x^n\right)=n{\log }_b(x)$
6. ${\log }_{b^k}(x)=\frac{1}{k}{\log }_b(x)$

These help in simplifying complex logarithmic expressions.

4.0Logarithmic Functions Rules

To work with logarithmic functions, apply these rules:

• Logs are only defined for positive values.
• The domain of $f(x)={\log }_b(x)isx>0$.
• The range is all real numbers.
• Use logarithmic identities to simplify, expand, or solve expressions.
• Convert between exponential and logarithmic forms to solve equations.

5.0Logarithmic Functions Graph

Graph of $f(x)={\log }_b(x)$:

• Passes through (1, 0)
• Has a vertical asymptote at x = 0
• Is increasing for b > 1, decreasing for 0 < b < 1
• Domain: x > 0
• Range: (−∞, ∞)

Example Graphs:

• $f(x)={\log }_2(x)$: Slowly increasing
• $f(x)=\ln (x)$: Natural logarithm curve
• $f(x)={\log }_{10}(x)$: Common logarithmic curve

6.0Logarithmic Functions Examples

Example 1: Convert the exponential equation $3^4=81$ into logarithmic form.

Solution: 

The logarithmic form is: ${\log }_3(81)=4$

Example 2: Evaluate ${\log }_{10}(1000)$

Solution: 

Since $10^3=1000$,

${\log }_{10}(1000)=3$

Example 3: Solve ${\log }_5(x)=2$

Solution:

Convert to exponential form:

$x=5^2=25$ 

Answer: x = 25

Example 4: Simplify ${\log }_2(8)+{\log }_2(4)$

Solution: 

Use the product rule:

${\log }_2(8\cdot 4)={\log }_2(32)$

Since $2^5=32$,

${\log }_2(32)=5$

Example 5: Evaluate ${\log }_2(10)$ using the change of base formula.

Solution:

${\log }_2(10)=\frac{{\log }_{10}(10)}{{\log }_{10}(2)}=\frac{1}{0.3010}\approx 3.32$

Example 6: Solve: ${\log }_2(x+2)=3$

Solution:

By definition,

$\begin{array}{rl}& x+2=2^3=8\\ & \Rightarrow x=8-2=6\end{array}$

Answer: x = 6

Example 7: Solve: ${\log }_{3}x+{\log }_3(x-2)=2$

Solution:
Using $\log a+\log b=\log (ab)$ :

$\begin{array}{rl}& {\log }_3[x(x-2)]=2\\ & \Rightarrow x(x-2)=3^2=9\\ & x^2-2x-9=0\\ & \Rightarrow x=1\pm \sqrt{10}\end{array}$ 

Only positive value allowed:

$x=1+10x=1+\sqrt{10}$.

Answer: $x=1+\sqrt{10}$

Example 8: Solve Logarithmic Inequality

Solve: $\ln x>2$

Solution:

$x>e^2$

Answer: $x>e^2$

Example 9: Evaluate: ${\lim }_{x\to 0^{+}}\frac{\ln (1+x)}{x}$ 

Solution:

Standard limit:

${\lim }_{x\to 0}\frac{\ln (1+x)}{x}=1$

Answer: 1

Example 10: Find: $\frac{d}{dx}\left(\ln \left(x^2+1\right)\right)$ 

Solution:

By chain rule:

$\frac{1}{x^2+1}\cdot 2x=\frac{2x}{x^2+1}$

Answer: $\frac{2x}{x^2+1}$

Example 11: Solve: $e^{\ln x}=5$ 

Solution:

$e^{\ln x}=x$. Thus,
x = 5.

Answer: x = 5

Example 12: Solve:

${\log }_5\left(x^2-1\right)={\log }_5(x-1)+{\log }_5(x+1)$

Solution:

RHS: 

$\begin{array}{rl}& {\log }_5(x-1)+{\log }_5(x+1)\\ & ={\log }_5[(x-1)(x+1)]\\ & ={\log }_5\left(x^2-1\right)\end{array}$

LHS = RHS → Identity valid.

Conditions:

$\begin{array}{rl}& x^2-1>0\Rightarrow |x|>1\text{ and }\\ & x-1>0\Rightarrow x>1\\ & x+1>0\Rightarrow x>-1\end{array}$

Common domain: x > 1

Answer: All x > 1

Example 13: Solve Inequality with Natural Logarithm

Solve:

$\ln \left(x^2+1\right)\le 2$

Solution:

$\begin{array}{rl}& x^2+1\le e^2\\ & \Rightarrow x^2\le e^2-1\\ & \Rightarrow |x|\le \sqrt{e^2-1}\end{array}$ 

Answer: $x\in \left[-\sqrt{e^2-1},\sqrt{e^2-1}\right]$

Example 14: Tangent to Logarithmic Curve

Find equation of tangent to $y=\ln xatx=1$.

Solution:
Point: (1, ln 1) = (1, 0)
Slope: $\frac{dy}{dx}=\frac{1}{x}\Rightarrow \frac{1}{1}=1$

Equation:

$\begin{array}{rl}& y-0=1\cdot (x-1)\\ & \Rightarrow y=x-1\end{array}$

Answer: y = x - 1

Example 15: Solve Mixed Equation with Logs

Solve:

${\log }_2(x-1)+{\log }_2(x+1)=3$

Solution:
Product rule:

$\begin{array}{rl}& {\log }_2[(x-1)(x+1)]=3\\ & \Rightarrow \left(x^2-1\right)=2^3=8\\ & \Rightarrow x^2=9\\ & \Rightarrow x=\pm 3\end{array}$

Check domain:

$x-1>0\Rightarrow x>1$ → Only x = 3 satisfies.

Answer: x = 3

6. What are some examples of logarithmic functions?

Examples include ${\log }_2(x),\ln (x),{\log }_{10}(x)$ and real-world applications like decibel scales and pH.

7. What are key logarithmic properties?

Product, quotient, and power rules:

• $\log (xy)=\log (x)+\log (y)$
• $\log \left(\frac{x}{y}\right)=\log (x)-\log (y)$
• $\log \left(x^n\right)=n\log (x)$

7.0Logarithmic and Exponential Functions

Logarithmic functions are inverses of exponential functions.

If:

$y=b^x\Rightarrow x={\log }_b(y)$

Thus, solving an exponential equation often requires applying logarithms, and vice versa.

8.0Practice Questions on Logarithmic Functions

1. Evaluate ${\log }_{10}(10000)$
2. Solve: ${\log }_3(x)=4$
3. Simplify: ${\log }_5(25)+{\log }_5(4)$
4. Use the change of base formula: ${\log }_4(32)$
5. Sketch the graph of $f(x)=\ln (x)$