Trouble with Logarithmic Equations: Help Needed for 5 Questions

[JBD2]

I'm having a lot of trouble on this worksheet I have, I've got most of the 32 questions except for about 5. I won't type out all the laws of logarithms as I assume that anyone coming in here to help me already knows them. So here are my questions (By the way, I've done work for them I just won't show it because it seems like I'm stuck where I'm at regardless):

17.) Solve for x:
$$(logx)^{logx}-9=0$$

24.) Solve for x, accurate to 2 decimal places
$$log_{4}x+log_{5}x=8.7$$

26.) Simplify completely.
$$\frac{log_{a}x}{log_{ab}x}-\frac{log_{a}x}{log_{b}x}$$

32.) Simplify/solve for x.
$$ln^{2}x+lnx^{3}+2=0$$

33.) Simplify/solve for x.
$$\frac{e^{x}+e^{-x}}{2}=k$$

Thank you for the help, I'm sorry about asking so many questions I just am really struggling on this one.

 

[JBD2]

Ok I did 26 and got this:

$$\frac{log_{a}x}{log_{ab}x}-\frac{log_{a}x}{log_{b}x}$$

$$\frac{logx}{loga}(\frac{logab}{logx})-\frac{logx}{loga}(\frac{logb}{logx})$$

$$\frac{logab}{loga} - \frac{logb}{loga}$$

$$log_{a}ab - log_{a}b$$

$$log_{a}\frac{ab}{b}$$

$$log_{a}a$$

= 1

So I think I got 26, is that right?

 

[statdad]

I haven't checked your calculations, but that is the right approach. You can use the same idea to solve 24 (convert the logs to the same base)

With a judicious use of a another logarithm rule you can convert 32 to a quadratic equation in $$\ln x$$

Multiplication by a correctly chosen exponential function (and clearing the fraction by multiplying everything by 2) will convert the final one to a quadratic equation in $$e^x$$

 

[JBD2]

Ok thank you, I have figured them all out now so it's good.