Question

Solve each equation. Show all your work. Round your answers to four decimal places.
4.) 74x = 415
17.) log7 (x +1) + log7(x – 5) = 1

I'm only stuck on these two problems, help please!!

Answer 1

4.) 7^4x = 415, sorry.

Answer 2

Okay, let's look at :
\[7^{4x}=415\]
Now, what we awant to do here, is create a exponent with the same base or but let's apply logarithms on both sides of the equality, ending with:
\[\log_{7} 7^{4x}=\log_{7} 415\]
Now, we have a property of the logarithms that I won't prove, but I'll show it to you:
\[\log_{}A ^{n} =n \log_{}A \]
The property states that if I have a logarithm number with an exponent, it's the same thing as making it a product with the exponent.
So let's apply that:
\[\log_{7} 7^{4x}=4x \log_{7} 7\]
But Logarithm with base 7 of 7 is equal to 1, meaning that:
\[4x \log_{7} 7=4x\]
\[\log_{7} 7=1\]
Let's replace what we found out on the original equation:
\[4x=\log_{7} 415\]
Now I want to leave the x alone on the left side of the equality, so I'll divide both sides by 4:
\[\frac{ 4x }{ 4 }=\frac{ \log_{7}415 }{ 4 }\]
now the 4's on the left side will simplify and the right side is equal to a constant.
\[x=\frac{ \log_{7} 415 }{ 4 }\]

Answer 3

so do i have to divide log7 415 and 4 to get x?

Answer 4

That's correct.

Answer 5

and what is it that log7 equals ?

Answer 6

\(\bf 7^{4x} = 415\implies log_7(7^{4x})=log_7415\implies 4x=log_7415\\ \quad \\
\textit{change of base rule of }{\color{blue}{ log_ab=\cfrac{logcb}{log_ca}}}\\
4x=log_7415\implies x=\cfrac{log_7415}{4}\implies
\large x=\cfrac{
{\color{blue}{ \frac{log_{10}415}{log_{10}7}}}
}{4}\)

Answer 7

ahemm \(\bf 7^{4x} = 415\qquad \textit{log cancellation rule of } {\color{blue}{ log_aa^x=x}}\) btw =)

Answer 8

There's no need for a change of base, Log7 (415) over 4, is already a constant.
We have no variables in game.

Answer 9

well.... true

Answer 10

But I believe you wanted to make it more, calculator friendly.
Thank you ~~

Answer 11

does it equal 7 because it cancels out?
7(415) / 4 = 726.25
x = 726.25?
im horrible with math.

Answer 12

https://www.google.com/search?client=opera&rls=en&q=(log(415)/log(7))/4&sourceid=opera&ie=utf-8&oe=utf-8&channel=suggest

Answer 13

\[x=\frac{ \log_{7}415 }{ 4 }\]
\[\log_{7} 415= 3.097922\]
\[=> x=\frac{ 3.097922 }{ 4 }\]
x=0.7744

Answer 14

as far as 17)
\(\bf recall\qquad {\color{blue}{ log(a)+log(b)\implies log(a\cdot b)}}\qquad thus\\ \quad \\
log_7(x +1) + log_7(x - 5) = 1\implies log_7[(x +1)(x - 5)]=1\\ \quad \\
\textit{log cancellation rule of }\Large {\color{red}{ a}}^{log_{\color{red}{ a}}x}=x\qquad thus\\ \quad \\

log_7[(x +1)(x - 5)]=1\implies\Large {\color{red}{ 7}}^{log_{\color{red}{ 7}}[(x +1)(x - 5)]}={\color{red}{ 7}}^1
\\ \quad \\
(x +1)(x - 5)=7\)

Answer 15

as you'd end up with a quadratic, set it to 0, factor it to get the values for "x"

Answer 16

ohhhhh, okay. thank you!!
and would you be able to help with the next one? (:

Answer 17

I'm about to dash, will be here tomorrow though :)