11.01 Exponential equations (Enrichment)
An exponential expression is an expression of the form $A^x$Ax, where $A$A is a positive number and $x$x is a pronumeral. $A$A is called the base of the exponential. An exponential equation is an equation where one or both sides are exponential expressions. To solve an exponential equation, we can write both sides of the equation as exponentials with the same base and then the indices will be equal.
Worked example
Solve $4^x=512$4x=512.
Think: Since $4^x$4x is an exponential expression, this is an exponential equation. In order to solve this equation, we will write both sides of the equation with the same base and then equate the indices.
Do: $4^x$4x is already an exponential, so how can we write $512$512 as an exponential? Notice that $512$512 is the result of multiplying together $2$2 nine times. That is, $512=2^9$512=29. Now we have $4^x=2^9$4x=29, so both sides are exponentials but they have different bases.
Can we write $4^x$4x with a base of $2$2? Notice that $4=2^2$4=22. Using index laws:
| $4^x$4x | $=$= | $\left(2^2\right)^x$(22)x |
Since $4=2^2$4=22 |
| $=$= | $2^{2x}$22x |
Using the rule $\left(A^m\right)^n=A^{mn}$(Am)n=Amn |
Now both sides of the equation have the same base.
| $4^x$4x | $=$= | $512$512 |
|
| $2^{2x}$22x | $=$= | $2^9$29 |
Since $4^x=2^{2x}$4x=22x and $512=2^9$512=29 |
| $2x$2x | $=$= | $9$9 |
Since the bases are the same we know that the indices are equal |
| $x$x | $=$= | $\frac{9}{2}$92 |
Dividing both sides of the equation by $2$2 |
So the solution is $x=\frac{9}{2}$x=92 which we can verify by substituting into the original equation.
Reflect: Notice that the original question is equivalent to asking what index do we raise $4$4 to to get $512$512. We could also solve this equation by noticing that $2=\sqrt{4}=4^{\frac{1}{2}}$2=√4=412 and $512=256\times2=4^4\times4^{\frac{1}{2}}=4^{\frac{9}{2}}$512=256×2=44×412=492. This gives the equation $4^x=4^{\frac{9}{2}}$4x=492, which gives us the same solution as above.
An exponential expression is an expression of the form $A^x$Ax, where $A$A is a positive number and $x$x is a pronumeral.
$A$A is called the base of the exponential.
An exponential equation is an equation where one or both sides are exponential expressions.
To solve an exponential equation, we can write both sides of the equation as exponentials with the same base and then the indices will be equal.
Practice questions
Question 1
Solve $3^x=3^6$3x=36.
Question 2
Solve $3^y=\frac{1}{27}$3y=127.
Question 3
Solve $9^x=\sqrt[9]{9}$9x=9√9.