Algebra II - Log-Base-10

Solve: $2log_{10}(100^3)$

$\textup{Cannot be determined.}$

Explanation

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

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

According to the log property:

$log_{b}(b^x) = x$

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

$2log_{10}(10^6) = 2(6) =12$

The answer is:

Solve the following:

$\log10000+\log10$

$\log10010$

Explanation

When the base isn't explicitly defined, the log is base 10. For our problem, the first term

$\log 10000=x$

is asking:

For the second term,

$\log(10)=y$

is asking:

So, our final answer is

Evaluate $\log_{8}172$ to the nearest tenth.

Explanation

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 ,

$\log_{M}N = \frac{\log_{a}N}{\log_{a}M}$

Setting , we can restate this logarithm as the quotient of two common logarithms, and calculate accordingly:

$\log_{8}172 = \frac{\log 172}{\log 8}\approx \frac{2.2355 }{0.9031}\approx 2.4754$

or, when rounded, 2.5.

This can also be done with natural logarithms, yielding the same result.

$\log_{8}172 = \frac{\ln 172}{\ln 8} \approx \frac{5.1475}{2.0794}\approx 2.4754$

Rond to one decimal place.

Evaluate: $2log_{10}(\frac{1}{1000})$

$\frac{2}{3}$

$\frac{1}{5}$

$\frac{1}{50}$

Explanation

We will need to write fraction in terms of the given base of log, which is ten.

$2log_{10}(\frac{1}{1000})=2log_{10}(10^{-3})$

According to the log rules:

$log_{x}(x^Y) =Y$

This means that the expression of log based 10 and the power can be simplified.

$2log_{10}(10^{-3}) =2\cdot -3 = -6$

The answer is:

Evaluate .

Explanation

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

$\log_ba=c \Rightarrow b^c=a$

$\log_{10}100=c \Rightarrow 10^c=100 \Rightarrow c=2$ .

Therefore

becomes,

$3\cdot2$ .

Finally, we do the simple multiplication:

Simplify: $log(10^{-6}\cdot 100^{-3})$

$\textup{No solution.}$

Explanation

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

Rewrite the inner quantity.

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

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

$log(10^{-12}) = log_{10}(10^{-12})$

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

$log_{x}x^y = y$

The answer is:

Solve:

Explanation

Break up using log rules. The log has a default base of ten.

$log(A\times B) = log(A)+log(B)$

$log(300) = log(3)+log(100)=log_{10}(3)+log_{10}(10^2)$

The exponent can be brought down as the coefficient since the bases of the second term are common.

$log_{10}(3)+log_{10}(10^2) =log_{10}(3)+2log_{10}(10) = log(3)+2$

This means that:

The answer is:

Evaluate: $2log(1000^{30})$

Explanation

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

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

Raise the coefficient of the log term as the power.

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

According to the log property:

$log_{b}(b^x) = x$

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

The answer is:

$7 ^{x} = 1,000$

Evaluate .

$x = \frac{3}{\log 7}$

$x = \frac{1,000}{\log 7}$

$x = 3 - \log 7$

$x = 1,000 - \log 7$

$\log 993$

Explanation

Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:

$7 ^{x} = 1,000$

$7 ^{x} = 10^{3}$

$\log 7 ^{x} = \log 10^{3}$

$x \log 7 = 3 \log 10$

$x \log 7 = 3 \cdot 1$

$x \log 7 = 3$

$\frac{x \log 7}{\log 7} = \frac{3}{\log 7}$

$x = \frac{3}{\log 7}$

Evaluate .

$5 ^{x} = 12$

$x=\frac{\log 12}{\log 5}$

$x= \log 7$

$x= \frac{ \log 12 }{5}$

$x=\frac{ \log 5}{\log 12}$

$x= \frac{ \log 5}{12}$

Explanation

Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:

$5 ^{x} = 12$

$\log 5 ^{x} = \log 12$

$x \log 5 = \log 12$

$\frac{x \log 5 }{\log 5 }=\frac{ \log 12}{\log 5}$

$x=\frac{ \log 12}{\log 5}$

Log-Base-10

Help Questions

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation