# Common and Natural Logarithms

## Logs with base 10 or base e, and the change of base formula.

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Common and Natural Logarithms

By now, you know that \begin{align*}log_2 64=x\end{align*} can be solved if you recognize that \begin{align*}2^{6} = 64\end{align*}. What about numbers that aren't so 'clean'? There aren't many people who could calculate the answer to \begin{align*}log_7 247=x\end{align*} in their head! It would be great to use a calculator, but most only have two log functions: base 10 and base e.

Is there a way to convert from one base to another, so we can use a calculator?

Embedded Video:

### Guidance

Although a log function can have any positive number as a base, there are really only two bases that are commonly used in the real world. Both may be written without a base noted, like: \begin{align*}log x\end{align*}, so you may need to use the context to decide which is appropriate.

The common log is a log with base 10. It is used to define pH, earthquake magnitude, and sound decibel levels, among many many other common real-world values.

The natural log, sometimes written \begin{align*}ln (x)\end{align*}, is a log with base e. The transcendental number e is approximately 2.71828 and is used in any number of calculations involving constant growth in chemistry, physics, biology, finance, etc..

Using a calculator for logs

You may have noticed that the common log and the natural log are the only log buttons on your calculator. We can use either the common log or the natural log to find the values of logs with other bases.

The equation \begin{align*}log_b x = \frac{log x} {log b}\end{align*} is called the change of base formula, and may be used to convert to common log or natural log.

You may also see the change of base formula as \begin{align*}\mathit\log_b x = \frac{ln x} {ln b}\end{align*}, which is the same formula specifying a conversion to the natural log.

Using the change of base formula, we can find the common log (or the natural log) equivalent of any other base so that we can use a calculator to find the value of an expression.

Consider log3 35. If we use the change of base formula to convert to base 10, and then the \begin{align*}log\end{align*} button on a calculator, we find that

\begin{align*}log_3 35 = \frac{log35} {log3} = 3.23621727\end{align*}.

#### Example A

Evaluate each log

a. \begin{align*}log 1\end{align*} b. \begin{align*}log 10\end{align*} c. \begin{align*}log\sqrt{10}\end{align*}

Solution:

Remember that \begin{align*}log x\end{align*} (with no base specified) commonly refers to \begin{align*}log_{10} x\end{align*}:

a. \begin{align*}log 1 = 0\end{align*} because \begin{align*}10^{0} = 1\end{align*}.
b. \begin{align*}log 10 = 1\end{align*} because \begin{align*}10^{1} = 10\end{align*}
c. \begin{align*}log \sqrt{10} = \frac{1} {2}\end{align*} because \begin{align*}\sqrt{10} = 10^{1/2}\end{align*}

#### Example B

For each log value, determine two integers between which the log value should lie. Then use a calculator to find the value of the log.

a. log 50 b. log 818

Solution:

a. log 50
The value of this log should be between 1 and 2, as 101 = 10, and 102 = 100.
Using a calculator, you should find that log 50 ≈ 1.698970004.
b. log 818
The value of this log should be between 2 and 3, as 102 = 100, and 103 = 1000.
Using a calculator, you should find that log 818 ≈ 2.912753304.

#### Example C

Estimate the value, and then use the change of base formula to find the value of \begin{align*}log_2 17\end{align*}.

Solution:

\begin{align*}log_2 17\end{align*} is close to 4 because \begin{align*}2^{4} = 16\end{align*} and \begin{align*}2^{5} = 32\end{align*}. Using the change of base formula, we have \begin{align*}log_2 17 = \frac{log 17} {log 2}\end{align*}. Using a calculator, you should find that the approximate value of this expression is 4.087462841.

Concept question wrap-up You should have no problem with the question from the beginning of the lesson now: \begin{align*}log_7 247=x\end{align*}.

Using the change of base formula: \begin{align*}log_7 247 = \frac{log 247} {log 7}\end{align*}

Using a calculator to find the common logs of 247 and 7, we get (approximately): \begin{align*}\frac{2.4} {.8} = 2.8313\end{align*}.

We can verify with: \begin{align*}7^{2.8313} = 247\end{align*}

\begin{align*}\therefore log_7 247 = 2.8313\end{align*}

### Vocabulary

Common Logarithm: A common logarithm is a log with base 10k. The log is usually written without the base.

Natural Logarithm: A natural log is a log with base e. The natural log is written as ln.

Trancendental Number: A number that is not the root of any rational polynomial function. Examples include \begin{align*}e\end{align*} and \begin{align*}\pi\end{align*}.

\begin{align*}e\end{align*} (constant): A constant used as the base of natural logarithms, equal to approximately 2.71828.

### Guided Practice

Questions

1) Find the value of each natural log.

a. \begin{align*}ln 100\end{align*} b. \begin{align*}ln \sqrt{e}\end{align*}

2) Solve the equation: \begin{align*}8^{x-3}=24\end{align*}

3) Solve the equation: \begin{align*}5^x = 3 \cdot 7^x\end{align*}

4) Find the value: \begin{align*}ln6 + ln7\end{align*}

5) Find the value: \begin{align*}log 5 - log 3\end{align*}

1) a. \begin{align*}ln 100\end{align*} is between 4 and 5. You can estimate this by rounding \begin{align*}e\end{align*} up to 3, and considering powers of 3:

\begin{align*}3^4 = 81\end{align*} and \begin{align*}3^5 = 243\end{align*}
Using a calculator, you should find that \begin{align*}ln 100\end{align*} ≈ 4.605171086.
b. Recall that a square root is the same as an exponent of 1/2. Therefore \begin{align*}ln \sqrt{e} = ln (e^{1/2}) = 1/2\end{align*}

2) To solve: \begin{align*}8^{x-3}=24\end{align*}:

\begin{align*}log 8^{x-3} = log 24\end{align*} : Take the log of both sides.
\begin{align*}(x-3) log 8 = log 24\end{align*} : Using \begin{align*}log x^y = y log x\end{align*}
\begin{align*}x = \frac{log 24}{log 8} +3\end{align*} : Divide both sides by \begin{align*}log 8\end{align*} and add \begin{align*}3\end{align*}.
\begin{align*}x = \frac{1.380}{.903} +3\end{align*} : With a calculator
\begin{align*}x = 4.528\end{align*} : With a calculator

3) To solve: \begin{align*}3^x(2^{3x}) = 7(5^x)\end{align*}:

\begin{align*}3^x(2^3)^x = 7(5^x)\end{align*} : Rule of exponents \begin{align*}(x^y)^z = x^{yz}\end{align*}
\begin{align*}3^x(8^x) = 7(5^x) \to 24^x = 7(5^x)\end{align*} : By multiplication
\begin{align*}(\frac{24}{5})^x = 7\end{align*} : Divide both sides by \begin{align*}5^x\end{align*}
\begin{align*}log (\frac{24}{5})^x = log 7\end{align*} : Take the log of both sides
\begin{align*}xlog (\frac{24}{5}) = log 7\end{align*} : Using \begin{align*}log x^y = y log x\end{align*}
\begin{align*}x = \frac{log7}{log\frac{24}{5}}\end{align*} : Divide both sides by \begin{align*}log (\frac{24}{5})\end{align*}
\begin{align*}x = 1.24\end{align*} : With a calculator

4) Use a calculator to find the values:

\begin{align*}ln6 = 1.79175\end{align*} and \begin{align*}ln7 = 1.94591\end{align*}
\begin{align*}1.79175 + 1.94591 = 3.73766\end{align*}

5) Recall that a \begin{align*}log\end{align*} without a specified base is commonly \begin{align*}log_{10}\end{align*}:

\begin{align*}log 5 = .69897\end{align*} and \begin{align*}log 3 = .477121\end{align*} : With a calculator
\begin{align*}.69897 - .477121 = .221849\end{align*}

### Practice

1. What is a common logarithm? Where are common logs most commonly used?
2. What is a natural logarithm? Where are natural logs commonly used?

Evaluate each expression:

1. \begin{align*}log\frac{17^4}{5}\end{align*}
2. \begin{align*}log 7(4^3)\end{align*}

Convert the Natural Logs to exponential form, and solve.

1. If \begin{align*}ln e = x\end{align*} and \begin{align*}e^x = e\end{align*} then \begin{align*}x =?\end{align*}
2. If \begin{align*}ln e^5\end{align*} then \begin{align*}x =?\end{align*}
3. If \begin{align*}ln e^a = x\end{align*} then \begin{align*}x =?\end{align*}
4. If \begin{align*}ln e^{-3} = x\end{align*} then \begin{align*}x =?\end{align*}

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