Define the term exponent .

An exponent is a power that a number (called the base ) is raised to.

What number do you get when you raise any non-zero number to the power of zero , e.g. 2 0 ?

Any non-zero number raised to the power of 0 is equal to 1 . E.g. 2 0 = 1.

True or False? Index laws only work with terms that have the same base .

True. Index laws only work with terms that have the same base .

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 .

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Exponentials & Logs (DP IB Applications & Interpretation (AI))

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Define the term exponent.
An exponent is a power that a number (called the base) is raised to.
What number do you get when you raise any non-zero number to the power of zero, e.g. 2⁰?
Any non-zero number raised to the power of 0 is equal to 1.
E.g. 2⁰ = 1.
True or False?
If you raise a non-zero number to the power of 1, you get 1.
False.
Any number raised to the power 1 is just itself.
E.g. $6^{1} = 6$ .
What do you get if you raise a non-zero number to the power of -1,
e.g. 3^-1?
If you raise a non-zero number to the power of -1 you get the reciprocal of the number.
E.g. $3^{- 1} = \frac{1}{3}$ .
What do you get if you raise a positive number to the power of ½,
e.g. 5^1/2?
If you raise a non-zero number to the power of ½ you get its positive square root.
E.g. $5^{\frac{1}{2}} = \sqrt{5}$ .
What is the index law for $x^{m} \times x^{n}$ ?
$x^{m} \times x^{n} = x^{m + n}$
If you multiply two powers with the same base number, you add the indices together.
This formula is not given in the exam formula booklet.
What is the index law for $x^{m} \div x^{n}$ ?
$x^{m} \div x^{n} = x^{m - n}$
If you divide two powers with the same base number, you subtract one index from the other.
This formula is not given in the exam formula booklet.
What is the index law for ${(x^{m})}^{n}$ ?
${(x^{m})}^{n} = x^{m \times n}$
If you raise a power to another power, you multiply the indices.
This formula is not given in the exam formula booklet.
What is the index law for ${(x y)}^{m}$ ?
${(x y)}^{m} = x^{m} y^{m}$
A power outside brackets is applied to each factor inside the brackets individually.
This formula is not given in the exam formula booklet.
But note that ${(x + y)}^{m} \neq x^{m} + y^{m}$ , i.e. you can only use this index law when the things inside the bracket are multiplied together.
Define the term reciprocal in relation to exponents.
In relation to exponents, the reciprocal of $x^{m}$ is $x^{- m}$ , which equals $\frac{1}{x^{m}}$ .
True or False?
Index laws only work with terms that have the same base.
True.
Index laws only work with terms that have the same base.
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.

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