Question

which of the following expressions are equal to 3? select all that apply. (2 points) $square$ $ln e^{3}$ $square$ $ln 3$ $square$ $log_{5} 15$ $square$ $log_{4} 64$ $square$ $log 1000$ $square$ $log 300$

Explanation:

Step1: Analyze $\ln e^{3}$

Using the property of logarithms $\ln a^{b}=b\ln a$ and $\ln e = 1$, we have $\ln e^{3}=3\ln e = 3\times1 = 3$.

Step2: Analyze $\ln 3$

$\ln 3$ is the natural logarithm of 3, and $\ln 3\approx1.0986
eq3$.

Step3: Analyze $\log_{5}15$

We know that $\log_{a}b=\frac{\ln b}{\ln a}$, so $\log_{5}15=\frac{\ln 15}{\ln 5}\approx\frac{2.70805}{1.60944}\approx1.6826
eq3$.

Step4: Analyze $\log_{4}64$

We know that if $\log_{a}x = y$, then $a^{y}=x$. For $\log_{4}64$, we need to find $y$ such that $4^{y}=64$. Since $4^{3}=64$, so $\log_{4}64 = 3$.

Step5: Analyze $\log 1000$ (assuming base 10)

If the base is 10, then $\log_{10}1000$ (since $\log$ without base is base 10) and we know that $10^{3}=1000$, so $\log 1000 = 3$.

Step6: Analyze $\log 300$ (assuming base 10)

$\log_{10}300=\log_{10}(3\times10^{2})=\log_{10}3 + \log_{10}10^{2}\approx0.4771+ 2=2.4771
eq3$.

Answer:

$\ln e^{3}$, $\log_{4}64$, $\log 1000$ (i.e., the options with $\ln e^{3}$, $\log_{4}64$, $\log 1000$)