1. ## Solving functions equation

Need help with the following.

Thanks

2. ## Re: Solving functions equation

a) $\displaystyle 25^{x-3}=5^x$

You can get the same base here since $\displaystyle 25=5^2$, so that you may write:

$\displaystyle (5^2)^{x-3}=5^x$

$\displaystyle 5^{2(x-3)}=5^x$

Now equate exponents, and solve for $\displaystyle x$.

b) $\displaystyle 2\left(\frac{1}{2} \right)^x-\left(\frac{1}{4} \right)^x=1$

Use the substitution $\displaystyle u=\left(\frac{1}{2} \right)^x$ so that you may nor write:

$\displaystyle 2u-u^2=1$

Now, solve the resulting quadratic, then back substitute for $\displaystyle u$.

c) $\displaystyle 3\ln(2x+5)=9$

Divide through by 3 to get:

$\displaystyle \ln(2x+5)=3$

Now convert from logarithmic to exponential form and solve for $\displaystyle x$.

d) $\displaystyle e^{(x-2)^2}=2$

Convert from exponential to logarithmic form:

$\displaystyle (x-2)^2=\ln(2)$

Now solve for $\displaystyle x$ using the square root property.

e) $\displaystyle 6^{-7x}=5$

Take the natural log of both sides to get:

$\displaystyle -7x\ln(6)=\ln(5)$

Now solve for $\displaystyle x$.