Question

assignment 3.5a
pre-calculus assignment: graphing log functions
graph the following. include 2 key points, and their coordinates. draw the vertical asymptote and an equation for the asymptote.

1. $f(x)=\log_3(x-2)$
2. $y=\log_2 x -1$
3. $y=3+\log_2 x$
4. $f(x)=\frac{1}{2}\ln x$
5. $f(x)=\frac{1}{2}\log(x-1)-2$
6. $f(x)=5-2\log(x+1)$

find each of the following without using a calculator.

1. $\ln e =$
2. $\log_5 1 =$
3. $\log_6 4 =$

convert to logarithmic form.

1. $e^3=t$
2. $8^t=x$
3. $e^x=7.3891$

convert to exponential form.

1. $\log_5 5=1$
2. $\log 7=0.845$

Explanation:

First Set: Evaluate Logarithms (No Calculator)

Step1: Use $\ln e=1$ property

$\ln e = 1$ (by definition, $\log_b b=1$)

Step2: Use $\log_b 1=0$ property

$\log_5 1 = 0$ (by definition, $\log_b 1=0$)

Step3: Rewrite $\log_6 4$ as change of base

$\log_6 4 = \frac{\ln 4}{\ln 6}$ (change of base formula $\log_b a=\frac{\ln a}{\ln b}$)

Step1: Exponential to logarithmic ($b^y=x\to\log_b x=y$)

For $e^3=t$: $\ln t = 3$

Step2: Exponential to logarithmic ($b^y=x\to\log_b x=y$)

For $8^t=x$: $\log_8 x = t$

Step3: Logarithmic to exponential ($\log_b x=y\to b^y=x$)

For $e^z=7.3891$: $\ln 7.3891 = z$

Step4: Logarithmic to exponential ($\log_b x=y\to b^y=x$)

For $\log_5 5=1$: $5^1 = 5$

Step5: Logarithmic to exponential ($\log_b x=y\to b^y=x$)

For $\log_7 7=0.845$: $7^{0.845} = 7$

For each function, identify the vertical asymptote (where the argument of the log is 0), find 2 key points by substituting $x$-values, and state the function equation.

Answer:

1. $1$
2. $0$
3. $\frac{\ln 4}{\ln 6}$

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Second Set: Convert Forms