Algebra II › Understanding Logarithms
Solve the following:
When the base isn't explicitly defined, the log is base 10. For our problem, the first term
is asking:
For the second term,
is asking:
So, our final answer is
Solve:
Change the base of the inner term or log to base ten.
According to the log property:
The log based ten and the ten to the power of will cancel, leaving just the power.
The answer is:
Solve:
Change the base of the inner term or log to base ten.
According to the log property:
The log based ten and the ten to the power of will cancel, leaving just the power.
The answer is:
Solve the following:
When the base isn't explicitly defined, the log is base 10. For our problem, the first term
is asking:
For the second term,
is asking:
So, our final answer is
Determine the value of:
The natural log has a default base of .
According to the rule of logs, we can use:
The coefficient in front of the natural log can be transferred as the power of the exponent.
The natural log and base e will cancel, leaving just the exponent.
The answer is:
Given the following:
Decide if the following expression is true or false:
for all positive
.
True
False
By definition of a logarithm,
if and only if
Take the th root of both sides, or, equivalently, raise both sides to the power of
, and apply the Power of a Power Property:
or
By definition, it follows that , so the statement is true.
, with
positive and not equal to 1.
Which of the following is true of for all such
?
By definition,
If and only if
Square both sides, and apply the Power of a Power Property to the left expression:
It follows that for all positive not equal to 1,
for all .
Evaluate to the nearest tenth.
Since most calculators only have common and natural logarithm keys, this can best be solved as follows:
By the Change of Base Property of Logarithms, if and
,
Setting , we can restate this logarithm as the quotient of two common logarithms, and calculate accordingly:
or, when rounded, 2.5.
This can also be done with natural logarithms, yielding the same result.
Rond to one decimal place.
Evaluate .
The first thing we can do is bring the exponent out of the log, to the front:
Next, we evaluate :
Recall that log without a specified base is base 10 thus
.
Therefore
becomes,
.
Finally, we do the simple multiplication:
Evaluate:
We will need to write fraction in terms of the given base of log, which is ten.
According to the log rules:
This means that the expression of log based 10 and the power can be simplified.
The answer is: