Exponential and Logarithmic Functions: Fresh Take

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Exponential and Logarithmic Functions: Fresh Take

Identify and evaluate exponential functions
Analyze logarithmic functions by identifying forms, understanding their exponential relationships, and calculating different base logarithms
Identify the hyperbolic functions, their graphs, and basic identities

Exponential Functions

The Main Idea

Definition of Exponential Functions:
- Form: $f(x) = ab^x$
- $a$ is a non-zero real number (initial value)
- $b$ is a positive real number, $b \neq 1$ (base)
Characteristics:
- Constant base, variable exponent
- Rapid growth compared to power functions
- Domain: all real numbers
- Range: $(0, ∞)$ for $b > 1$ , $(0, ∞)$ for $0 < b < 1$
Evaluating Exponential Functions:
- Substitute the given value for $x$
- Follow order of operations
- Pay attention to parentheses and exponents
Laws of Exponents:
- Product of Powers: $b^x \cdot b^y = b^{x+y}$
- Quotient of Powers: $\frac{b^x}{b^y} = b^{x-y}$
- Power of a Power: $(b^x)^y=b^{xy}$
- Power of a Product: $(ab)^x=a^x b^x$
- Power of a Quotient: $\dfrac{a^x}{b^x} =\left(\dfrac{a}{b}\right)^x$
Applications:
- Population growth
- Compound interest
- Radioactive decay

Given the exponential function $f(x)=100·3^{x/2}$ , evaluate $f(4)$ and $f(10)$ .

$f(4)=900; \, f(10)=24,300$ .

Go to World Population Balance for another example of exponential population growth.

Simplify the following expression and then evaluate it for $x = 2$ :

$\dfrac{(3x^{2})^3 \cdot 2^{x-1}}{9x^{-1} \cdot 4^{x+2}}$

Simplify using laws of exponents:

$\begin{array}{rcl} \dfrac{(3x^{2})^3 \cdot 2^{x-1}}{9x^{-1} \cdot 4^{x+2}} &= & \dfrac{3^3 \cdot (x^2)^3 \cdot 2^{x-1}}{3^2 \cdot x^{-1} \cdot (2^2)^{x+2}} \\ &=& \dfrac{27x^6 \cdot 2^{x-1}}{9x^{-1} \cdot 2^{2(x+2)}} \\ &=& \dfrac{27x^6 \cdot 2^{x-1}}{9x^{-1} \cdot 2^{2x+4}} \\ &=& \dfrac{3x^7 \cdot 2^{x-1}}{2^{2x+4}} \\ &=& 3x^7 \cdot 2^{x-1-(2x+4)} \\ &=& 3x^7 \cdot 2^{-x-5} \end{array}$

Evaluate for $x = 2$ :

$3(2^7) \cdot 2^{-2-5} = 3 \cdot 128 \cdot 2^{-7} = 384 \cdot \frac{1}{128} = 3$

Use the laws of exponents to simplify $\dfrac{(6x^{-3}y^2)}{(12x^{-4}y^5)}$ .

$x^a/x^b=x^{a-b}$

$\dfrac{x}{(2y^3)}$

Logarithmic Functions

The Main Idea

Definition of Logarithmic Functions:
- Inverse of exponential functions
- $\log_b(x) = y$ if and only if $b^y = x$
- Domain: $(0, ∞)$ , Range: $(-∞, ∞)$
Common Logarithms:
- Base 10: $\log_{10}(x)$ , often written as $\log(x)$
- Natural log: $\log_e(x)$ , written as $\ln(x)$
Properties of Logarithms:
- Product: $\log_b(xy) = \log_b(x) + \log_b(y)$
- Quotient: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
- Power: $\log_b(x^n) = n\log_b(x)$
Change-of-Base Formula:
- $\log_a(x) = \dfrac{\log_b(x)}{\log_b(a)}$
- Useful for calculating logs with non-standard bases
Solving Logarithmic Equations:
- Often involves converting to exponential form
- May require using logarithm properties to simplify

Solve $\dfrac{e^{2x}}{(3+e^{2x})}=\dfrac{1}{2}$ .

First solve the equation for $e^{2x}$ .

$x=\dfrac{\ln 3}{2}$

Watch the following video to see the worked solution to this example.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “1.5 Exponential and Logarithmic Functions” here (opens in new window).

Solve $\ln(x^3)-4 \ln (x)=1$ .

$x=\frac{1}{e}$

Use the change-of-base formula and a calculator to evaluate $\log_4 6$ .

$1.29248$

Hyperbolic Functions

The Main Idea

Definitions of Hyperbolic Functions:
- $\cosh x = \dfrac{e^x + e^{-x}}{2}$
- $\sinh x = \dfrac{e^x - e^{-x}}{2}$
- $\tanh x = \dfrac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
- $\text{csch} x = \dfrac{1}{\sinh x}$
- $\text{sech} x = \dfrac{1}{\cosh x}$
- $\coth x = \dfrac{\cosh x}{\sinh x}$
Graphs and Behavior:
- $\cosh x$ : Similar to $|e^x|$ , always $≥ 1$
- $\sinh x$ : Odd function, similar to $e^x$ for large x
- $\tanh x$ : Odd function, approaches $±1$ as $x → ±∞$
Key Identities:
- $\cosh^2 x - \sinh^2 x = 1$
- $1 - \tanh^2 x = \text{sech} ^2 x$
- $\coth^2 x-1=\text{csch}^2 x$
- $\sinh(x \pm y)=\sinh x \cosh y \pm \cosh x \sinh y$
- $\cosh (x \pm y)=\cosh x \cosh y \pm \sinh x \sinh y$
- $\cosh x + \sinh x = e^x$
- $\cosh x-\sinh x=e^{−x}$
Inverse Hyperbolic Functions:
- $\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1})$
- $\cosh^{-1} x = \ln(x + \sqrt{x^2 - 1})$ ( $x ≥ 1$ )
- $\tanh^{-1} x = \frac{1}{2}\ln(\frac{1+x}{1-x})$ ( $|x| < 1$ )

Simplify and evaluate $\sinh(\cosh^{-1}(3))$ .

Start with the inside function:

$\cosh^{-1}(3) = \ln(3 + \sqrt{3^2 - 1}) = \ln(3 + \sqrt{8}) = \ln(3 + 2\sqrt{2})$

Now we have $\sinh(\ln(3 + 2\sqrt{2}))$ . Recall that $\sinh(\ln x) = \frac{x - \frac{1}{x}}{2}$

Apply this:

$\sinh(\ln(3 + 2\sqrt{2})) = \dfrac{(3 + 2\sqrt{2}) - \dfrac{1}{3 + 2\sqrt{2}}}{2}$

Simplify the denominator:

$\dfrac{1}{3 + 2\sqrt{2}} = \dfrac{3 - 2\sqrt{2}}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} = \dfrac{3 - 2\sqrt{2}}{9 - 8} = 3 - 2\sqrt{2}$

Substitute back:

$\dfrac{(3 + 2\sqrt{2}) - (3 - 2\sqrt{2})}{2} = \dfrac{4\sqrt{2}}{2} = 2\sqrt{2}$

Simplify $\cosh(2 \ln x)$ .

$\frac{(x^2+x^{-2})}{2}$

Evaluate $\tanh^{-1}\left(\frac{1}{2}\right)$ .

Use the definition of $\tanh^{-1} x$ and simplify.

$\frac{1}{2}\ln(3) \approx 0.5493$ .