1.6 Logarithmic Functions and Equations - Introduction to Calculus

Logarithmic Functions

Logarithms are the inverse of exponential functions. Let $a > 0$ and $a \ne 1$ , and let $x > 0$ ; then we define a logarithm (log) as follows:

THEOREM

Logarithmic Functions

\begin{aligned} y & = \log_{\scriptscriptstyle a}(x) \\[0.65em] a^y & = x \end{aligned}

If no base $a$ is shown, a base of 10 is assumed.

For example:

\log(x) = \log_{\scriptscriptstyle 10}(x)

Since any positive number to the power of 0 is equal to 1, we have the property that $\log_{\scriptscriptstyle a}(1) = 0$ , no matter what the base $a$ is.

Move the base and input of $\displaystyle y=\log_{\scriptscriptstyle a}(x)$ . The second curve uses the reciprocal base. Compare their values at the same input.

\displaystyle y=\log_{\scriptscriptstyle 2}(x)

\displaystyle y=\log_{\scriptscriptstyle \frac{1}{2}}(x)

Base a2Input x1

\displaystyle \log_{\scriptscriptstyle 2}(1)=0

\displaystyle 2^{0}\approx 1

Increasing · Domain: x > 0

Try a = 2, then a = 0.5. What changes at a = 1? At x = 1, every valid logarithm has value zero.

Domain: $0 < x < \infty$
Range: $y \in \mathbb{R}$
$y = \log_{\scriptscriptstyle a}(x)$ always passes through $(a, 1)$ and $(1, 0)$
If $a > 1$ then the graph of $y = \log_{\scriptscriptstyle a}(x)$ is increasing.
If $0 < a < 1$ then the graph of $y = \log_{\scriptscriptstyle a}(x)$ is decreasing.
$x = 0$ is always a vertical asymptote of $y = \log_{\scriptscriptstyle a}(x)$

QUESTION

Which base is invalid for a real logarithm?

Evaluate each of the following.

(a) $\displaystyle \log_{\scriptscriptstyle 2} 8$

(b) $\displaystyle \log(100)$

(d) $\displaystyle \log_{\scriptscriptstyle 8} 1$

Logarithm Rules

THEOREM

Logarithm Rules

\begin{aligned} & \log_{\scriptscriptstyle a}(a^x) = x \\[0.65em] & a^{\log_{\scriptscriptstyle a}(x)} = x \\[0.65em] & \log_{\scriptscriptstyle a}(xy) = \log_{\scriptscriptstyle a}(x) + \log_{\scriptscriptstyle a}(y) \\[0.65em] & \log_{\scriptscriptstyle a}\left({\dfrac{x}{y}}\right) = \log_{\scriptscriptstyle a}(x) - \log_{\scriptscriptstyle a}(y) \\[0.65em] & \log_{\scriptscriptstyle a}(x^n) = n\log_{\scriptscriptstyle a}(x) \end{aligned}

Change of Base Formula

We can switch between any two bases easily by using the formula:

THEOREM

Change of Base Formula

\begin{aligned} \log_{\scriptscriptstyle a}(x) = {\dfrac{\log_{\scriptscriptstyle b}(x)}{\log_{\scriptscriptstyle b}(a)}} \end{aligned}

For these logarithm rules, $x,y>0$ and $a,b>0$ with $a,b\ne1$ In the power rule, $n$ may be real.

QUESTION

Which expression equals

\displaystyle \log_{\scriptscriptstyle 4}(x)

, for

\displaystyle x>0

Prove the change of base formula.

For $\displaystyle x>0$ , $\displaystyle a,b>0$ , and $\displaystyle a,b\ne1$ , show that $\displaystyle \displaystyle \log_{\scriptscriptstyle a}(x)=\dfrac{\log_{\scriptscriptstyle b}(x)}{\log_{\scriptscriptstyle b}(a)}$

Convert $\displaystyle \log_{\scriptscriptstyle 4}(x)$ into a logarithm with each of the following bases.

(a) base 3

(b) base 22

The Natural Logarithm

A special logarithm is the natural logarithm, which is the logarithm with a base of $e$ Rather than write $\log_{\scriptscriptstyle e}(x)$ , we typically write $\ln(x)$

The natural logarithm has the exact same properties as any other logarithmic function.

THEOREM

Natural Logarithm

\begin{aligned} & \ln(e^x) = x \\[0.65em] & e^{\ln(x)} = x \end{aligned}

QUESTION

What are all real solutions of

\displaystyle \log_{\scriptscriptstyle 2}(x^4)+\log_{\scriptscriptstyle 2}(x^2)=6

Solve each of the following for $\displaystyle x$

(a) $\displaystyle 2^x=2^{1-x}$

(b) $\displaystyle 3^{\frac12x+10}=27$

(d) $\displaystyle \log(x)-1=\log(x-1)$

(e) $\displaystyle \log_{\scriptscriptstyle 2}(x)+\log_{\scriptscriptstyle 2}(x^2)=6$

(f) $\displaystyle \log_{\scriptscriptstyle 2}(x^4)+\log_{\scriptscriptstyle 2}(x^2)=6$

Logarithmic Functions

Logarithms are the inverse of exponential functions. Let $a > 0$ and $a \ne 1$ , and let $x > 0$ ; then we define a logarithm (log) as follows:

THEOREM

Logarithmic Functions

\begin{aligned} y & = \log_{\scriptscriptstyle a}(x) \\[0.65em] a^y & = x \end{aligned}

If no base $a$ is shown, a base of 10 is assumed.

For example:

\log(x) = \log_{\scriptscriptstyle 10}(x)

Since any positive number to the power of 0 is equal to 1, we have the property that $\log_{\scriptscriptstyle a}(1) = 0$ , no matter what the base $a$ is.

Move the base and input of $\displaystyle y=\log_{\scriptscriptstyle a}(x)$ . The second curve uses the reciprocal base. Compare their values at the same input.

\displaystyle y=\log_{\scriptscriptstyle 2}(x)

\displaystyle y=\log_{\scriptscriptstyle \frac{1}{2}}(x)

Base a2Input x1

\displaystyle \log_{\scriptscriptstyle 2}(1)=0

\displaystyle 2^{0}\approx 1

Increasing · Domain: x > 0

Try a = 2, then a = 0.5. What changes at a = 1? At x = 1, every valid logarithm has value zero.

Domain: $0 < x < \infty$
Range: $y \in \mathbb{R}$
$y = \log_{\scriptscriptstyle a}(x)$ always passes through $(a, 1)$ and $(1, 0)$
If $a > 1$ then the graph of $y = \log_{\scriptscriptstyle a}(x)$ is increasing.
If $0 < a < 1$ then the graph of $y = \log_{\scriptscriptstyle a}(x)$ is decreasing.
$x = 0$ is always a vertical asymptote of $y = \log_{\scriptscriptstyle a}(x)$

QUESTION

Which base is invalid for a real logarithm?

Evaluate each of the following.

(a) $\displaystyle \log_{\scriptscriptstyle 2} 8$

(b) $\displaystyle \log(100)$

(d) $\displaystyle \log_{\scriptscriptstyle 8} 1$

Logarithm Rules

THEOREM

Logarithm Rules

\begin{aligned} & \log_{\scriptscriptstyle a}(a^x) = x \\[0.65em] & a^{\log_{\scriptscriptstyle a}(x)} = x \\[0.65em] & \log_{\scriptscriptstyle a}(xy) = \log_{\scriptscriptstyle a}(x) + \log_{\scriptscriptstyle a}(y) \\[0.65em] & \log_{\scriptscriptstyle a}\left({\dfrac{x}{y}}\right) = \log_{\scriptscriptstyle a}(x) - \log_{\scriptscriptstyle a}(y) \\[0.65em] & \log_{\scriptscriptstyle a}(x^n) = n\log_{\scriptscriptstyle a}(x) \end{aligned}

Change of Base Formula

We can switch between any two bases easily by using the formula:

THEOREM

Change of Base Formula

\begin{aligned} \log_{\scriptscriptstyle a}(x) = {\dfrac{\log_{\scriptscriptstyle b}(x)}{\log_{\scriptscriptstyle b}(a)}} \end{aligned}

For these logarithm rules, $x,y>0$ and $a,b>0$ with $a,b\ne1$ In the power rule, $n$ may be real.

QUESTION

Which expression equals

\displaystyle \log_{\scriptscriptstyle 4}(x)

, for

\displaystyle x>0

Prove the change of base formula.

Convert $\displaystyle \log_{\scriptscriptstyle 4}(x)$ into a logarithm with each of the following bases.

(a) base 3

(b) base 22

The Natural Logarithm

A special logarithm is the natural logarithm, which is the logarithm with a base of $e$ Rather than write $\log_{\scriptscriptstyle e}(x)$ , we typically write $\ln(x)$

The natural logarithm has the exact same properties as any other logarithmic function.

THEOREM

Natural Logarithm

\begin{aligned} & \ln(e^x) = x \\[0.65em] & e^{\ln(x)} = x \end{aligned}

QUESTION

What are all real solutions of

\displaystyle \log_{\scriptscriptstyle 2}(x^4)+\log_{\scriptscriptstyle 2}(x^2)=6

Solve each of the following for $\displaystyle x$

(a) $\displaystyle 2^x=2^{1-x}$

(b) $\displaystyle 3^{\frac12x+10}=27$

(d) $\displaystyle \log(x)-1=\log(x-1)$

(e) $\displaystyle \log_{\scriptscriptstyle 2}(x)+\log_{\scriptscriptstyle 2}(x^2)=6$

(f) $\displaystyle \log_{\scriptscriptstyle 2}(x^4)+\log_{\scriptscriptstyle 2}(x^2)=6$

Logarithmic Functions and Equations

Logarithmic Functions

Logarithmic Functions

Explore logarithms and their bases

KNOWLEDGE QUIZ

Evaluate logarithms

Logarithm Rules

Logarithm Rules

Change of Base Formula

Change of Base Formula

KNOWLEDGE QUIZ

Prove the change of base formula

Change logarithm bases

The Natural Logarithm

Natural Logarithm

KNOWLEDGE QUIZ

Solve exponential and logarithmic equations

Logarithmic Functions and Equations

Logarithmic Functions

Logarithmic Functions

Explore logarithms and their bases

KNOWLEDGE QUIZ

Evaluate logarithms

Logarithm Rules

Logarithm Rules

Change of Base Formula

Change of Base Formula

KNOWLEDGE QUIZ

Prove the change of base formula

Change logarithm bases

The Natural Logarithm

Natural Logarithm

KNOWLEDGE QUIZ

Solve exponential and logarithmic equations