Help with logarithmic equation?

Question

Find the value of x if:

$(2^x)^2 + 3(2^x) - 18 = 0$

So far, I have done

$(2^x)^2 + 2^x(3)=18$

$(2^x)^2+2^x=6$

What should i do with $(2^x)^2+2^x$ so i can have only one $^X$ on the left side of the equation?

Let $t=2^x$, and solve the quadratic. Then $x=\log_2t$, where $t>0$. — Lucian, Mar 09 '14 at 12:21

Yiyuan Lee · Accepted Answer · 2014-03-09T12:29:55.820

3

Let $A = 2^x$. Then the equation becomes

$$A^2 + 3A - 18 = 0$$

which is just one of your everyday quadratic equations. Now, solve the quadratic equation for $A$ to find that $A = -6, 3$. Equate both with $2^t$ to find that

$$2^t = -6, 3$$

For the case when $2^t = 3$, we have $2^t = 3 \implies t = \log_2{3}$.

We reject the case for which $2^t = -6$, because $2^t$ is always positive for real $t$.

edited Mar 09 '14 at 12:29

answered Mar 09 '14 at 12:24

Yiyuan Lee

14,435

Help with logarithmic equation?

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