little trouble on logarithms?

Question

1) solve and round to 2 decimal places:
4*(5^x)-3*(0.4^2x)=11

2)write as a single logrithm, please give me a few steps before taking me to the final answer, thanks!
(4*log[5](x) - 2*log[5](y)) / (3*log[5](w))

Answer 1

2) Make use of the facts that
log (a) - log(b) = (log/ab),
b log (a) = log(a^b), and
log [base a](b) / log [base a] (c) = log [base c] (b)

( 4 log [5] (x) - 2 log[5](y) ) / 3log[5](w)
( log [5] (x^4) - log[5](y^2) ) / log[5](w^3)
log [5](x^4 / y^2) / log[5](w^3)
log [base w^3] (x^4 / y^2)

Answer 2

I responded to question #2 more then 40 minutes ago. What's wrong with the solution:
(4*log[5](x) - 2*log[5](y)) / (3*log[5](w)) =
(log[5](x^4) - log[5](y^2)) / (log[5](w^3))
log [5] (x^4/y^2) / log[5] (w^3) =
log[w^3] (x^4/y^2)

You have only 87 points. Try responding some questions. It's your turn

Answer 3

4*(5^x)-3*(0.4^2x)=11
log4 + xlog5 -(log 3+2xlog.4)=log 11
log 4 +xlog5 -log3 -2xlog.4 =log11
x(log5 -2log.4) = log 11 -log4 +log3
x = (log 11 -log4 +log3)/(log5 -2log.4)
x = .61

(4*log[5](x) - 2*log[5](y)) / (3*log[5](w))
Let's assume everything is log to base 5
(log x^4 -logy^2)/logw^3
=log(x^4/y^2)/logw^3

Answer 4

1) x = 0.61

Solution

4(5^x) - 3(0.4^2x) = 11

Take logs

ln[4(5^x)] - ln[3(0.4^2x)] = ln11

Separate multiplicative logs

ln4 + ln(5^x) - ln3 - ln(0.4^2x) = ln11

ln4 + xln(5) - ln3 - 2xln0.4 = ln11

x(ln5 - 2ln0.4) = ln11 - ln4 + ln3

x = (ln11 - ln4 + ln3)/(ln5 - 2ln0.4)

x = 0.61

(Hope thats right, may have been a bit sloppy)

2) the way you've written it shows a common factor of ln5, so I'm not sure what you mean

Answer 5

1) 5^x - 3*(0.4^2x) = 11

Rule 1 => log (base a) x = y <=> x = a^y

Rule 2 => log(base b) x = (log(bas a) x)/log(base a) b



=> log (base 5) x = log x/log 5

=> log(base 0.4) 2x = log 2x/log 0.4


(log 0.4) (log x) -3log2 -3logx -3log5 = 11(log 5) (log 0.4)

Put y = log x

(log 0.4) y -3 log2 -3y -3log5 = 11(log5)(log 0.4)

y[log(0.4) -3] -3log2 = 3log5 + 11(log5)(log 0.4)

y = 3log2 + log5 + 11(log5) log(0.4)
--------------------------------------------
log(0.4) -3


log x = 3log2 + log5 + 11(log5) log(0.4)
---------------------------------------------
log(0.4) - 3


10^log x = 10 ^ same algebraic expression



x = 10 ^ (3log2 + log5 + (11log5)(log 0.4)/log(0.4) -3