Properties of Logarithms

Help Questions

Pre-Calculus › Properties of Logarithms

Questions 1 - 10

Condense the logarithm

Explanation

In order to condense the logarithmic expression, we use the following properties

As such

Completely expand this logarithm:

$\ln \frac{\sqrt{6x-2}}{8}$

$\frac{1}{2}\ln(6x-2)-\ln 8$

$\frac{1}{2}\ln(6x) -\ln (2)-\ln 8$

$\ln(6x-2)^{\frac{1}{2}}-\ln 8$

$\ln(6x-2)^{\frac{1}{2}}-2.08$

The answer is not present.

Explanation

We expand logarithms using the same rules that we use to condense them.

Here we will use the quotient property

$\ln\frac{u}{v}= \ln u-\ln v$

and the power property

$\ln u^{n}=n\ln u$ .

$\ln \frac{\sqrt{6x-2}}{8}$

Use the quotient property:

$\ln \sqrt{6x-2}-\ln 8$

Rewrite the radical:

$\ln (6x-2)^{\frac{1}{2}}-\ln 8$

Now use the power property:

$\frac{1}{2}\ln (6x-2)-\ln 8$

Completely expand this logarithm:

$\ln \frac{\sqrt{6x-2}}{8}$

$\frac{1}{2}\ln(6x-2)-\ln 8$

$\frac{1}{2}\ln(6x) -\ln (2)-\ln 8$

$\ln(6x-2)^{\frac{1}{2}}-\ln 8$

$\ln(6x-2)^{\frac{1}{2}}-2.08$

The answer is not present.

Explanation

We expand logarithms using the same rules that we use to condense them.

Here we will use the quotient property

$\ln\frac{u}{v}= \ln u-\ln v$

and the power property

$\ln u^{n}=n\ln u$ .

$\ln \frac{\sqrt{6x-2}}{8}$

Use the quotient property:

$\ln \sqrt{6x-2}-\ln 8$

Rewrite the radical:

$\ln (6x-2)^{\frac{1}{2}}-\ln 8$

Now use the power property:

$\frac{1}{2}\ln (6x-2)-\ln 8$

Condense the logarithm

Explanation

In order to condense the logarithmic expression, we use the following properties

As such

Condense the following logarithmic equation:

$3\log (y)+\log (4)+\log (x)-2\log (z)$

$\log \left(\frac{4xy^3}{z^2}\right)$

$\log \left(\frac{z^2}{4xy^3}\right)$

$\log \left(\frac{xy^3}{2z}\right)$

$\log (4xy^3-z^2)$

$\log \left(\frac{2z^2}{xy^3}\right)$

Explanation

We start condensing our expression using the following property, which allows us to express the coefficients of two of our terms as exponents:

$b\log (a)=\log (a^b)$

$3\log (y)+\log (4)+\log (x)-2\log (z)$

$\log(y^3)+\log(4)+\log(x)-\log(z^2)$

Our next step is to use the following property to combine our first three terms:

$\log (a)+\log(b)+\log(c)=\log(abc)$

$\log(y^3)+\log(4)+\log(x)-\log(z^2)$

$\log(4xy^3)-\log(z^2)$

Finally, we can use the following property regarding subtraction of logarithms to obtain the condensed expression:

$\log (a)-\log (b)=\log (\frac{a}{b})$

$\log(4xy^3)-\log(z^2)$

$\log \left(\frac{4xy^3}{z^2}\right)$

Condense the following logarithmic equation:

$3\log (y)+\log (4)+\log (x)-2\log (z)$

$\log \left(\frac{4xy^3}{z^2}\right)$

$\log \left(\frac{z^2}{4xy^3}\right)$

$\log \left(\frac{xy^3}{2z}\right)$

$\log (4xy^3-z^2)$

$\log \left(\frac{2z^2}{xy^3}\right)$

Explanation

We start condensing our expression using the following property, which allows us to express the coefficients of two of our terms as exponents:

$b\log (a)=\log (a^b)$

$3\log (y)+\log (4)+\log (x)-2\log (z)$

$\log(y^3)+\log(4)+\log(x)-\log(z^2)$

Our next step is to use the following property to combine our first three terms:

$\log (a)+\log(b)+\log(c)=\log(abc)$

$\log(y^3)+\log(4)+\log(x)-\log(z^2)$

$\log(4xy^3)-\log(z^2)$

Finally, we can use the following property regarding subtraction of logarithms to obtain the condensed expression:

$\log (a)-\log (b)=\log (\frac{a}{b})$

$\log(4xy^3)-\log(z^2)$

$\log \left(\frac{4xy^3}{z^2}\right)$

Solve the following logarithmic equation:

$2\log _3(x)=\log _3(4)+\log _3(x-1)$

Explanation

In order to solve this equation, we must apply several properties of logarithms. First we notice the term on the left side of the equation, which we can rewrite using the following property:

$a\log_b (x)=\log_b (x^a)$

Where a is the coefficient of the logarithm and b is some arbitrary base. Next we look at the right side of the equation, which we can rewrite using the following property for the addition of logarithms:

$\log _ba+\log _bc=\log_b (a\cdot c)$