Pre-Calculus - Logarithmic Functions

Evaluate the following logarithmic expression:

$\log _5\left(\frac{1}{25}\right)+\log _7(49)$

$\log \left(\frac{7}5{}\right)$

$\log \left(\frac{49}{25}\right)$

Explanation

In order to evaluate the logarithmic expression, we have to remember the notation of a logarithm and what it means:

$\log _ab$

Any logarithm expressed in the form above is simply asking "a to what power equals b?" So we're trying to find what power the base must be raised to in order to obtain the value in parentheses. If we look at our expression in particular:

$\log _5\left(\frac{1}{25}\right)+\log _7(49)$

The first term is asking "5 to what power equals 1/25?" while the second is asking "7 to what power equals 49?" Setting these questions up mathematically, we can find the values of each logarithm:

$5^x=\frac{1}{25}\rightarrow 5^x=\frac{1}{5^2}\rightarrow 5^x=5^{-2}\rightarrow x=-2$

$7^x=49\rightarrow 7^x=7^2\rightarrow x=2$

So the value of the first term is -2, and the value of the second term is 2, which gives us:

$\log _5\left(\frac{1}{25}\right)+\log _7(49)=-2+2=0$

Evaluate:

$\log_2256$

Explanation

To evaluate, you must convert 256 to a power of 2. Then the log and 2 will cancel to leave you with your answer.

$\log_22^8=8$

Evaluate the following logarithm:

$\log_5125$

Explanation

The notation of this logarithm is asking "5 to what power equals 125?" If we set this up mathematically, we can simplify either side until it is readily apparent what the value of x is, which is the value of the logarithm:

So to answer our question, 5 to the power of 3 is 125, so the answer is 3.

Which of the following diagrams represents the graph of the following logarithmic function?

$y=log_{5}x$

Varsity log graph

Explanation

For $y=log_{5}x$ , is the exponent of base 5 and is the product. Therefore, when , $x=5^{1}$ and when , $x=5^{3 }$ . As a result, the correct graph will have values of 5 and 125 at and , respectively.

Evaluate the following logarithm: $\log _5(7)$

$x=\frac{\log (7)}{\log (5)}$

$x=\frac{\log (5)}{\log (7)}$

$x=\frac{\log (7)}{\log (10)}$

$x=\frac{\log (10)}{\log (5)}$

$x=\frac{\log (5)}{\log (10)}$

Explanation

The problem asks us to evaluate the following logarithm:

$\log _5(7)$

According to the definition of a logarithm, what this equation is asking us is "5 to what power equals 7?":

Using the properties of logarithms, if we take the log of both sides we get the following equation and simplification:

$\log (5^x)=\log (7)$

$x\log (5)=\log (7)$

$x=\frac{\log (7)}{\log (5)}$

Which of these is a correct method for evaluating $log_{4}8$ ?

$\frac{log_24}{log_28}$

Explanation

When evaluating a log you are asking what power your base must be raised by to get your argument.

If you cannot directly figure that out you can use a change of base formula instead.

$log_ab=\frac{log_cb}{log_ca}$

Evaluate to four decimal places: $\log _{5} 7$

Explanation

Using a calculator, evaluate $\frac{\log7}{\log5} \approx 1.2091$

Evaluate this logarithm: $f(x)=\log_{3}x,\hspace{1mm}x=243$

$f(243)=\log_{3}243=5$

$f(243)=\log_{3}243=6$

$f(243)=\log_{3}243=3$

$f(243)=\log_{3}243=8$

$f(243)=\log_{3}243=3$

Explanation

It is important to remember a logarithm is really just an exponent. In fact when you see $\log_{a}x$ remember that the expression is only asking what exponent must "a" be raised to in order to obtain "x"!