Question

1. solve $4^{x}=36$. write your answer as a single logarithm.

$x =$

1. a. write $12^{x}=85$ as a single logarithm.

b. using a calculator, find the value of $x$ to the nearest ten-thousandth.
$x \approx$

1. a. write $\log_{6}24$ as a common logarithm.

b. using a calculator, find the value of $\log_{6}24$ to the nearest ten-thousandth.
$x \approx$

1. solve the following equation for $x$.

$7 = -2 + 3\log_{2}x$
$x =$

1. rewrite the following exponential equations in logarithmic form.

a. $e^{8}=5x$
b. $e^{5}=8x$
c. $e^{5x}=8$
d. $e^{10x}=5$

Explanation:

Step1: Convert to logarithm form

For $4^x=36$, use $\log_b(a)=x$ for $b^x=a$.
$x=\log_{4}36$

Step2: Rewrite $12^x=85$ as logarithm

Apply logarithmic conversion rule.
$x=\log_{12}85$

Step3: Calculate $\log_{12}85$ value

Use change of base: $\log_{12}85=\frac{\log(85)}{\log(12)}$
$\frac{\log(85)}{\log(12)}\approx1.8761$

Step4: Rewrite $\log_6 24$ as common log

Apply change of base formula.
$\log_{6}24=\frac{\log(24)}{\log(6)}$

Step5: Calculate $\log_6 24$ value

Compute the ratio of common logs.
$\frac{\log(24)}{\log(6)}\approx1.7737$

Step6: Isolate the logarithmic term

For $7=-2+3\log_2 x$, add 2 to both sides.
$7+2=3\log_2 x \implies 9=3\log_2 x$

Step7: Solve for the logarithm

Divide both sides by 3.
$\frac{9}{3}=\log_2 x \implies 3=\log_2 x$

Step8: Convert to exponential form

Rewrite $\log_2 x=3$ to solve for $x$.
$x=2^3=8$

Step9: Rewrite $e^8=5x$ to log form

Use $\ln(a)=b$ for $e^b=a$.
$\ln(5x)=8$

Step10: Rewrite $e^5=8x$ to log form

Apply natural log conversion rule.
$\ln(8x)=5$

Step11: Rewrite $e^{5x}=8$ to log form

Map exponential to natural logarithm.
$\ln(8)=5x$

Step12: Rewrite $e^{6x}=5$ to log form

Use natural log for base-$e$ exponential.
$\ln(5)=6x$

Answer:

1. $\log_{4}36$
2. a. $\log_{12}85$

b. $1.8761$

1. a. $\frac{\log(24)}{\log(6)}$

b. $1.7737$

1. $8$
2. a. $\ln(5x)=8$

b. $\ln(8x)=5$
c. $\ln(8)=5x$
d. $\ln(5)=6x$