Convert between logarithms and exponentials
Convert between exponential and logarithmic forms
Let a,b,c be real numbers with a \gt 0. Then:
a^b=c\Leftrightarrow b=\log_a\left(c\right)
Consider the following equation:
3^4=81
We have:
4=\log_3\left(81\right)
Solve equations and inequalities involving exponentials and logarithms
Solve equations of the form a ^ x = b \text{ with } b \gt 0
How to solve a^x=b
Suppose a,b are positive real numbers. Consider the following equation:
a^x=b
The solution to the equation is:
x=\log_a\left(b\right)
Consider the following equation:
3^x=7
The solution to the equation is:
x=\log_3\left(7\right)
Solve inequalities of the form a^x \gt b
How to solve a^x \gt b
Suppose a,b are real numbers. Consider the following inequality:
a^x \gt b
The solution set to the inequality is:
x \gt \log_a\left(b\right)
Consider the following inequality:
2^x \gt 4
The solution set to the inequality is:
x \gt \log_2\left(4\right)=2
Solve equations of the form x^a=b
How to solve x^a=b
Suppose that a is a positive integer and b is a real number. Consider the following equation:
x^a=b
The solution to the equation is:
\begin{cases} \pm\sqrt[a]{b}& \mbox{ if }b\geq0\mbox{ and }a \mbox{ is even} \cr \cr \mbox{no solutions} & \mbox{if }b \lt 0 \mbox{ and }a\mbox{ is even}\cr \cr \sqrt[a]{b}&\mbox{ if }a\mbox{ is odd} \end{cases}
Consider the following equation:
x^2=49
The solution to the equation is:
x=\pm \left(49\right)^{1/2}=\pm \sqrt{49}
Therefore, there are two solutions to the equation:
x=7 and x=-7
The two solutions can also be found by graphing y=x^2 and y=49.
The x -coordinates of the two intercepts correspond to the solutions of the equation.
Consider the following equation:
x^2=-49
There are no solutions to the equation.
We can also see that there are no solutions to the equation graphically. If we graph y=x^2 and y=-49, then we see that there are no intercepts. Therefore, there are no solutions to the equation.
Consider the following equation:
x^3=27
The solution to the equation is:
x=\sqrt[3]{27}=3
The solution to the equation can be found by graphing y=x^3 and y=27.
The x -coordinate of the intersection of the two lines is the solution to the equation.
Solving inequalities of the form x^a \gt b
How to solve inequalities of the form x^a \gt b
Suppose a is a positive integer and b is a real number. Consider the following equation:
x^a \gt b
The solution to the inequality is:
\begin{cases} \left(-\infty,-\sqrt[a]{b}\right)\cup \left(\sqrt[a]{b},\infty\right)& \mbox{ if }b\geq0\mbox{ and }a \mbox{ is even} \cr \cr \left(-\infty,\infty\right) & \mbox{if }b \lt 0 \mbox{ and }a\mbox{ is even}\cr \cr \left(\sqrt[a]{b},\infty\right)&\mbox{ if }a\mbox{ is odd} \end{cases}
Consider the following inequality:
x^2 \gt 16
Then \sqrt{16}=4 and the solution set to the inequality is:
\left(-\infty,-4\right)\cup \left(4,\infty\right)
The solution set can be seen by graphing y=x^2 and y=16. The x -values where the graph of y=x^2 lies above the line y=16 correspond to solutions to the inequality.
Observe that the graph of y=x^2 lies above the line y=16 to the left of the intersection point \left(-4{,}16\right) and to the right of the intersection point \left(4{,}16\right).
Consider the following inequality:
x^2 \gt -16
The solution set to the inequality is:
\left(-\infty,\infty\right)
The solution set can be seen by graphing y=x^2 and y=-16. The x -values where the graph of y=x^2 lies above the line y=16 correspond to solutions to the inequality.
Observe that the graph of y=x^2 lies everywhere above the line y=-16.
Consider the following inequality:
x^3 \gt -8
Then \sqrt[3]{-8}=-2 and the solution set to the inequality is:
\left(-2,\infty\right)
The solution set can be seen by graphing y=x^3 and y=-8. The x -values where the graph of y=x^3 lies above the line y=-8 correspond to solutions to the inequality.
Observe that the graph of y=x^3 lies above the line y=-8 to the right of the intersection point \left(-2,-8\right).