# How do you solve 62^(x+3) <= 7^(2x+1)?

Mar 19, 2016

take the common logarithm of both sides and you will get:

$\log {62}^{x + 3} \le \log {7}^{2 x + 1}$

using the power rule for logs, the exponents become factors or multipliers and the equation reduces to:

$\left(x + 3\right) \log 62 \le \left(2 x + 1\right) \log 7$
by expanding , we will obtain a simple linear equation with 1 variable (x):

xlog62 + 3log62 $\le$ 2xlog7 + 1log7

collect all your x terms on one side of the equation and the others on the opposite side:

xlog62 - 2xlog7 $\le$ log7 - 3log62

factoring the common factor of 'x':

x(log62 - 2log7) $\le$ log7 - 3log62

and the result is:

x $\le$ $\frac{\log 7 - 3 \log 62}{\log 62 - 2 \log 7}$

Mar 19, 2016

$x \text{ ">=" } \frac{\ln \left({7}^{5}\right)}{\ln \left({7}^{2} / 62\right)} - 3$

If you prefer: $x \text{ ">=" } \frac{5 \ln \left(7\right)}{2 \ln \left(7\right) - \ln \left(62\right)} - 3$

#### Explanation:

You could use any form of log for this. I chose ${\log}_{e}$

Taking logs of both sides

$\text{ } \left(x + 3\right) \ln \left(62\right) \textcolor{red}{\le} \left(2 x + 1\right) \ln \left(7\right)$

$\text{ } \implies \frac{\ln \left(62\right)}{\ln \left(7\right)} \textcolor{red}{\le} \frac{2 x + 1}{x + 3}$

Dividing the right hand side gives

$\text{ } \frac{\ln \left(62\right)}{\ln \left(7\right)} \textcolor{red}{\le} 2 - \frac{5}{x + 3}$

$\text{ } \frac{\ln \left(62\right)}{\ln \left(7\right)} - 2 \textcolor{red}{\le} - \frac{5}{x + 3}$

Multiply by (-1)

$\text{ } 2 - \frac{\ln \left(62\right)}{\ln \left(7\right)} \textcolor{red}{\ge} + \frac{5}{x + 3}$

$\text{ } \left(x + 3\right) \left(2 - \frac{\ln \left(62\right)}{\ln \left(7\right)}\right) \textcolor{red}{\ge} + 5$

$\text{ } x + 3 \textcolor{red}{\ge} 5 \times \frac{\ln \left(7\right)}{2 \ln \left(7\right) - \ln \left(62\right)}$

$\text{ } x \textcolor{red}{\ge} \frac{5 \ln \left(7\right)}{2 \ln \left(7\right) - \ln \left(62\right)} - 3$

$\text{ } x \textcolor{red}{\ge} \frac{\ln \left({7}^{5}\right)}{\ln \left({7}^{2} / 62\right)} - 3$