Question

I need help solving: solve terms in natural logarithms log4 x + log 11 x=1

Answer 1

\[\large \log_4x+\log_{11}x=1\]So is this what the problem looks like? and they want us tooooooo convert to natural log to find x? :o

Answer 2

yes

Answer 3

We'll use the change of base formula,\[\large \log_ab=\frac{\log_cb}{\log_ca}\]

Where c is any value we want.
In this problem we want c=e. The log of base e is the natural log.

Answer 4

So for the first term, applying this rule gives us,\[\large \log_4x= \frac{\log_e x}{\log_e 4}\]Which we can of course simply write as,\[\large \frac{\ln x}{\ln 4}\]

Answer 5

So we'll get something similar for the next term,
giving us an equation of,\[\large \frac{\ln x}{\ln4}+\frac{\ln x}{\ln11}=1\]

Answer 6

So the wording at the start is a little confusing.
Do they want us to solve for \(x\)? or can we stop when we get to \(\ln x\)?

Answer 7

Solve for x and decimal approximation

Answer 8

From our last step, we'll factor ln x out of each term on the left,\[\large \ln x\left(\frac{1}{\ln4}+\frac{1}{\ln11}\right)=1\]
If we get a common denominator inside the brackets, we'll have,\[\large \ln x\left(\frac{\ln11+\ln4}{\ln4\cdot\ln11}\right)=1\]Multiplying both sides by the RECIPROCAL of what's in the brackets gives us,\[\large \ln x = \left(\frac{\ln4\cdot\ln11}{\ln4+\ln11}\right)\]

From here, we can exponentiate both sides:
Rewrite both sides with a base e,\[\huge e^{\ln x} = e^{\left(\frac{\ln4\cdot\ln11}{\ln4+\ln11}\right)}\]

Answer 9

On the left, the exponential and logarithm are inverse operations of one another, so they essentially cancel out.\[\huge x = e^{\left(\frac{\ln4\cdot\ln11}{\ln4+\ln11}\right)}\]

Answer 10

Then we just have to punch this into the calculator to solve for x.

It's a bit tough to get through all the steps :( I know.

Answer 11

i got 5.75 is that right?

Answer 12

Hmmm no. Maybe having a little calculator trouble there.

Answer 13

oh wait 7.80

Answer 14

4.80 oops

Answer 15

\[\large \text{e ^}((\ln(4)\ln(11))/(\ln(4)+\ln(11))\]

It depends how you're putting it into the calculator.
If you want to put it all in at once, it will look like this.
It's a lot of brackets so it can be a bit confusing :)

Answer 16

The number you're trying to get issssss,
2.40714491

So try to get that out of your calculator c:

Answer 17

got it...I switched calc's .....cell phone calculator was doing it ...thank you!