# Exponents and Logs

#### nuckers

Why is the 5 in this equation, I can't multiply the 2 and 5, some of this math is so confusing

$$\displaystyle 2(5)^x=7^{x+1}$$

#### e^(i*pi)

MHF Hall of Honor
Why is the 5 in this equation, I can't multiply the 2 and 5, some of this math is so confusing

$$\displaystyle 2(5)^x=7^{x+1}$$
The 5 is the base of the exponential. You can still take logs and use the relevant log laws.

Hint: $$\displaystyle \log(2\cdot 5^x) = \log(2) + \log(5^x)$$

#### Plato

MHF Helper
$$\displaystyle 2(5)^x=7^{x+1}$$
That equation is equivalent to $$\displaystyle \left( {\frac{5}{7}} \right)^x = \frac{7}{2}$$ which has solution $$\displaystyle x = \frac{{\log (7) - \log (2)}}{{\log (5) - \log (7)}}$$

#### nuckers

That equation is equivalent to $$\displaystyle \left( {\frac{5}{7}} \right)^x = \frac{7}{2}$$ which has solution $$\displaystyle x = \frac{{\log (7) - \log (2)}}{{\log (5) - \log (7)}}$$
I'm totally confused how you came up with that. I did screw up and put a 7 in the equation, it's supposed to be $$\displaystyle 3^{x+1}$$

#### Plato

MHF Helper
I'm totally confused how you came up with that. I did screw up and put a 7 in the equation, it's supposed to be $$\displaystyle 3^{x+1}$$
$$\displaystyle 2 \cdot 5^x = 3^{x + 1}$$ is equivalent to $$\displaystyle \left( {\frac{5}{3}} \right)^x = \frac{3}{2}$$ which has solution $$\displaystyle x = \frac{{\log (3) - \log (2)}}{{\log (5) - \log (3)}}$$
To see how that works divide both sides by $$\displaystyle 3^x$$ and also by $$\displaystyle 2$$.
Note that $$\displaystyle \frac{{5^x }}{{3^x }} = \left( {\frac{5}{3}} \right)^x$$.

#### nuckers

$$\displaystyle 2 \cdot 5^x = 3^{x + 1}$$ is equivalent to $$\displaystyle \left( {\frac{5}{3}} \right)^x = \frac{3}{2}$$ which has solution $$\displaystyle x = \frac{{\log (3) - \log (2)}}{{\log (5) - \log (3)}}$$
To see how that works divide both sides by $$\displaystyle 3^x$$ and also by $$\displaystyle 2$$.
Note that $$\displaystyle \frac{{5^x }}{{3^x }} = \left( {\frac{5}{3}} \right)^x$$.
Ok, i understand it now, thanks so much for the help, i appreciate it