In order to eliminate the log based ten, we will need to raise both sides as the exponents using the base of ten.

The equation becomes:

\(\displaystyle 10^{log_{10}(2x)} =10^{-2}\)

The ten and log based ten will cancel, leaving just the power on the left side.  Change the negative exponent into a fraction on the right side.

\(\displaystyle 2x=\frac{1}{100}\)

Divide by two on both sides, which is similar to multiplying by a half on both sides.

\(\displaystyle 2x\cdot \frac{1}{2}=\frac{1}{100}\cdot \frac{1}{2}\)

Simplify both sides.

The answer is:  \(\displaystyle x=\frac{1}{200}\)

In order to solve this, we will need to rewrite the base \(\displaystyle 100\) as \(\displaystyle 10^2\), since log is by default base 10.

Rewrite the expression.

\(\displaystyle log_{10}(10^{2(100)})=log_{10}(10^{200})\)

By log rules, the exponent can be pulled down as the coefficient.

\(\displaystyle 200log_{10}(10) = 200(1) = 200\)

The answer is:  \(\displaystyle 200\)

One of the properties of logs is the ability to cancel out terms based on the base of the log. Since the base of the log is 10 we can simplify the 100 to 10 squared.

\(\displaystyle log_{10}100=log_{10}10^2\)

The log base 10 and the 10 cancel out, leaving you with the value of the exponent, 2 as the answer.

The log term has a default base of 10.  The 1000 will need to be rewritten as base 10.

\(\displaystyle 2log(1000^{30}) =2 log_{10}(10^{3(30)})\)

Raise the coefficient of the log term as the power.

\(\displaystyle log_{10}(10^{3(30)(2)}) = log_{10}(10^{180})\)

According to the log property:

\(\displaystyle log_{b}(b^x) = x\)

The log based 10 and the 10 inside the quantity of the log will cancel, leaving just the power.

The answer is:  \(\displaystyle 180\)

The log has a default of base ten.  This means we should convert the 1000 to a common base 10.

\(\displaystyle 1000= 10^3\)

Replace this value inside the log term.

\(\displaystyle log(1000^{500}) = log_{10}(10^{3(500)})\)

Since the log base 10 and the ten to a certain power are existent, they will both cancel, leaving just the power itself.

\(\displaystyle log_{b}(b^x) = x\)

\(\displaystyle 3(500) = 1500\)

The answer is:  \(\displaystyle 1500\)

Change the base of the inner term or log to base ten.

\(\displaystyle 2log_{10}(100^3)= 2log_{10}(10^{2(3)})= 2log_{10}(10^6)\)

According to the log property:

\(\displaystyle log_{b}(b^x) = x\)

The log based ten and the ten to the power of will cancel, leaving just the power.

\(\displaystyle 2log_{10}(10^6) = 2(6) =12\)

The answer is:  \(\displaystyle 12\)

Rewrite the log such that it is in its simplest form.  Break up the 500 with common factors.

\(\displaystyle log(500) = log(2\times 2\times 5\times 5\times 5\)\)

This can be broken into addition of logs.

\(\displaystyle log(2)+log(2)+log(5)+log(5)+log(5)\)

\(\displaystyle = 2log(2)+3log(5)\)

The answer is:  \(\displaystyle 2log(2)+3log(5)\)

When the inner terms of a log are divided, we can simply rewrite separate logs using subtraction.

\(\displaystyle log(\frac{3}{1000})= log(3)-log(1000)\)

Note that log has a default base of ten, and we can rewrite the 1000 as ten to the power of three.

\(\displaystyle log(3)-log(1000) = log(3)-log(10^3)\)

Use the property \(\displaystyle log_xx^y=y\) to simplify the second term.

\(\displaystyle log(10^3)=3\)

The answer is:  \(\displaystyle log(3)-3\)

The log is in default base 10.  To simplify this log, we will need to change the base of 100 to base 10.

\(\displaystyle 100=10^2\)

Rewrite the inner quantity.

\(\displaystyle log(10^{-6}\cdot 100^{-3})=log(10^{-6}\cdot 10^{2(-3)})=log(10^{-6}\cdot 10^{-6} )\)

We can use the additive rule of exponents since both bases are the same.

\(\displaystyle log(10^{-12}) = log_{10}(10^{-12})\)

According to the rule of logs, a log of a base with similar bases will cancel, and will leave only the power.

\(\displaystyle log_{x}x^y = y\)

The answer is:  \(\displaystyle -12\)

In order to eliminate the log, which has a default base of 10, we will need to raise both sides of the equation as powers using the value of 10.

\(\displaystyle 10^{log_{10}(3x)}= 10^3\)

The equation becomes:

\(\displaystyle 3x= 1000\)

Divide by three on both sides.

\(\displaystyle \frac{3x}{3}=\frac{ 1000}{3}\)

The answer is:  \(\displaystyle \frac{ 1000}{3}\)