What is a logarithm ?

A logarithm is the inverse of an exponent .

True or False? You can take a log of a negative number.

False. You can not take a log of a negative number.

True or False? The laws of logarithms can only be used for logarithms with the same base .

True. The laws of logarithms can only be used for logarithms with the same base .

True or False? Logarithms can be used to solve exponential equations .

True. Logarithms can be used to solve exponential equations .

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Exponentials & Logs (DP IB Analysis & Approaches (AA))

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FrontIntroduction to Logarithms
What is a logarithm?

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What is a logarithm?
A logarithm is the inverse of an exponent.
State the logarithm equation in terms of $a$ , $b$ , and $x$ that is equivalent to $a^{x} = b$ .
If $a^{x} = b$ , then the equivalent logarithm equation is $\log_{a} b = x$ .
This is valid so long as $a > 0$ , $b > 0$ and $a \neq 1$ .
This logarithm equation is given in the exam formula booklet.
What is $\ln x$ the notation for?
$\ln x$ is the notation for the natural logarithm of $x$ .
This is equivalent to $\log_{e} x$ , where $e$ is the mathematical constant approximately equal to 2.718.
True or False?
$\log x$ is sometimes used as an abbreviation for $\log_{10} x$ .
True.
$\log x$ is sometimes used as an abbreviation for $\log_{10} x$ .
$\log x$ will usually mean $\log_{10} x$ , unless otherwise specified.
Define the term base in the context of logarithms.
In the context of logarithms, the base is the number that is being raised to a power in the equivalent exponential equation.
E.g. in $\log_{a} b$ the base is $a$ .
True or False?
The equation $2^{x} = 10$ can be solved using logarithms.
True.
The equation $2^{x} = 10$ can be solved using logarithms, specifically by finding the value of the solution $x = \log_{2} 10$ .
What is the logarithm law for $\log_{a} x y$ ?
$\log_{a} x y = \log_{a} x + \log_{a} y$
If you take the log of the product of two numbers, it is the same as the sum of the log of each number.
The logs of both individual numbers must have the same base.
This formula is given in the exam formula booklet.
What is the logarithm law for $\log_{a} \frac{x}{y}$ ?
$\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$
If you take the log of the division of two numbers, it is the same as the difference of the log of each number.
The logs of both individual numbers must have the same base.
This formula is given in the exam formula booklet.
What is the logarithm law for $\log_{a} x^{m}$ ?
$\log_{a} x^{m} = m \log_{a} x$
If you take the log of a number raised to the power of another number, it is the same as the product of the power and the log of the number.
This formula is given in the exam formula booklet.
What is the result of $\log_{a} 1$ , given $a > 0, a \neq 1$ ?
$\log_{a} 1 = 0$ , given $a > 0, a \neq 1$ .
The log of 1, for any positive base that is not equal to 1, is always 0.
This is equivalent to $a^{0} = 1$ .
This result is not in your exam formula booklet.
True or False?
$\log_{a} a = 0$ .
False.
$\log_{a} a = 1$ .
The log of a number, where the base of the log is the same as the number, is always equal to 1.
This result is not in your exam formula booklet.
What is the result of $\log_{a} (a^{x})$ ?
$\log_{a} (a^{x}) = x$
This is the result of the logarithm law $\log_{a} x^{m} = m \log_{a} x$ and the fact that $\log_{a} a = 1$ .
This also illustrates the fact that logarithms and exponents (with the same base) are inverses.
This result is not in your exam formula booklet.
True or False?
$\log_{a} x^{n} = {(\log_{a} x)}^{n}$
False.
$\log_{a} x^{n} \neq {(\log_{a} x)}^{n}$
$\log_{a} x^{n} = n \log_{a} x$
True or False?
$e^{\ln x} = x$
True.
$e^{\ln x} = x$
Also, $a^{\log a} = a$ , the logarithm and the exponent 'cancel' each other out.
These results are not in your exam formula booklet.
How can the expression $\ln e^{x}$ be simplified?
$\ln e^{x} = x$ .
The natural log and the exponent 'cancel' each other out.
True or False?
You can take a log of a negative number.
False.
You can not take a log of a negative number.
True or False?
The laws of logarithms can only be used for logarithms with the same base.
True.
The laws of logarithms can only be used for logarithms with the same base.
What is the change of base logarithm law?
The change of base logarithm law allows you to change the base of a logarithm using the formula
$\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$
This is given in your exam formula booklet.
True or False?
Logarithms can be used to solve exponential equations.
True.
Logarithms can be used to solve exponential equations.
What is the first step required to solve a basic exponential equation, e.g. $e^{x} = 12$ ?
The first step required to solve a basic exponential equation is to take logarithms of both sides.
E.g. for $e^{x} = 12$ , take the natural log of both sides
The equation then becomes $x \ln e = \ln 12$ , which simplifies to $x = \ln 12$ (because $\ln e = 1$ ).
True or False?
You can use the change of base law, $\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$ , to solve some exponential equations.
True.
You can use the change of base law, $\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$ , to solve some exponential equations.
Some questions may require you to change the base of a particular logarithm in order to apply other logarithm laws.

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