Exponential and Logarithmic Functions

Review basic rules of exponential and logarithmic functions
Published

August 24, 2024


1 Exponential function

  • The exponential function has the form \(\displaystyle y= f(x) =a^x\):

    • \(a\) is a constant known as the base, \(a \gt 0\) and \(a \ne 0\).

    • \(x\) is the exponent.

    • The natural exponential function \(y = e^x\) is a specific type of exponential functions where the base is the constant \(e\), known as Euler’s number (\(e \approx 2.718281828459045…\)).

    • A graphical presentation of an exponential function with bases less or greater than \(1\) is depicted in Figure 1.

Figure 1: Plot of exponential functions with bases less or greater than 1

2 Logarithmic function

  • The logarithmic function has the form \(y = f(x) = \log_ax\):

    • \(a\) is a constant known as the base of logarithm, \(a \gt 0\) and \(a \ne 1\).

    • \(x \gt 0\) (i.e., the function is defined only if \(x\) is positive).

    • It is read as \(\log\) to the base \(a\) of \(x\).

    • If the base \(a=10\), then it is known as the common logarithm (\(\log\)), but if \(a = e\), then it is known as the natural logarithm (\(\ln\)).

    • R function for the common logarithm is log10() and for the natural logarithm is log().

    • A graphical presentation of the logarithmic function \(y =log_{10}\ x\) is depicted in Figure 2.

Figure 2: Plot of logarithmic function with base 10

3 Relation between exponential and logarithmic functions

  • Each function is the inverse of the other (i.e., each function undoes the other):

    • \(a^{\log_ax} = x\), for \(x>0\)

    • \(\log_a a^x = x\)

  • If \(a^y = x\), then \(y = \log_a x\).

  • Similarly, if \(\log_a x = y\), then \(x = a^y\).

4 Basic laws of exponents

  1. \(a^m a^n = a^{m+n}\)

  2. \(\displaystyle \frac{a^m}{a^n} = a^{m-n}\)

  3. \((a^m)^n = a^{mn}\)

  4. \((ab)^n = a^nb^n\)

  5. \(\displaystyle (\frac{a}{b})^n = \frac{a^n}{b^n}\)

  6. \(a^0 = 1\)

  7. \(\displaystyle a^{-n} = \frac{1}{a^n}\)

  8. \(\displaystyle a^{\frac{1}{n}} =\sqrt[n]{a}\), this is referred to as the \(n^{th}\) root of \(a\)

  9. \(\displaystyle a^{\frac{m}{n}} = \sqrt[n]{a^m}\)

Calculate \(\displaystyle \frac{2^7}{2^5} + 4^0\)

\[ \displaystyle \frac{2^7}{2^5} + 4^0 = \] \[ \displaystyle 2^{(7-5)} + 4^0\ {\color{#0466c8}(\text {apply rule 2}\rightarrow)} = \] \[ \displaystyle 2^2 + 1\ {\color{#0466c8}(\text {apply rule 6}\rightarrow)} = \] \[ \displaystyle 4 + 1 = 5 \]

5 Basic laws of logarithms

  1. \(\log_a x^b = b\log_a x\)

  2. \(\log_a (xy) = \log_a x + \log_a y\)

  3. \(\log_a \frac{x}{y} = \log_a x - \log_a y\)

  4. \(\log_a x = \displaystyle \frac{\log_b x}{\log_b a}\)

  5. \(\log_a x = \displaystyle \frac{1}{\log_x a}\)

  6. \(\log_a 1 = 0\)

  7. \(\log_a a = 1\)

Write \((4 \log_{10} x + 3 \log_{10} 5y)\) as a single logarithm.

\[ 4 \log_{10} x + 3 \log_{10} 5y = \]

\[ \log_{10} x^4 + \log_{10} (5y)^3 {\color{#0466c8}(\text {apply rule 1}\rightarrow)} = \] \[ \log_{10} x^4 \cdot (5y)^3 {\color{#0466c8}(\text {apply rule 2}\rightarrow)} = \] \[ \log_{10} 125 x^4 y^3 \]

Note

To apply the above manipulations, the base of the logarithm should be the same for all terms.

Solve the equation \(125^{x-2} = 5^{2-x}\) for \(x\)

\[ 125^{x-2} = 5^{2-x} \Rightarrow \] \[ (5^3)^{x-2} = 5^{2-x} \Rightarrow \] \[ 5^{3 (x-2)} = 5^{2-x} \Rightarrow \] \[ 5^{3x-6} = 5^{2-x} \Rightarrow \] \[ 3x-6 = 2-x \Rightarrow \] \[ 3x+x = 2+6 \Rightarrow \] \[ 4x = 8 \Rightarrow \] \[ x = 2 \]

Solve the equation \(\log_2 (3y-1) = 3\) for \(y\)

The answer is \(y =\)

If \(y = \log_a x\), then \(a^y = x\)

Solve the equation \(3\ln (\sqrt[3]{y-e}) - \ln 2 = 1\) for \(y\)

The answer is \(y =\)

6 References


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