Need help with logs?

Question

I've looked through my book and also went in for help but this stuff isn't clicking with me. I already failed the class once, and would rather not do so again. I guess resorting to this shows that at least I care.

In advance, I'd like to thank anyone who takes the time to assist me. I appreciate it.

I'll use the letter v to symbolize a down arrow.

1) Use the properties of logarithms to rewrite this as a single logarithm.

4 logV10^x + 2 logV10^(x+3)

2) Solve for x

logV7^(x+3) - 2 = logV7^3

3) Solve for x

4^x-3 = 25

4) Approx to 2 decimal places

logV25^750

Answer 1

[1]
4 log10 (x) + 2 log10 (x + 3)
A coefficient in front of a log is the same as an exponent for the log.
log10 (x^4) + log10 (x + 3)²
The sum of logs is equal to the log of the product.
log10 [ x^4 · (x + 3)²]
If you like, you could multiply this out to get:
log10 (x^6 + 6x^5 + 9x^4)

[2]
log7 (x + 3) - 2 = log7 (3)
Use a base of 7 and raise to the exponents of your logs.
7^[log7 (x + 3) - 2] = 7^[log7 (3)]
(x + 3) / 49 = 3
x + 3 = 147
x = 144

[3]
4^(x - 3) = 25
Use log (base 4) on both sides.
x - 3 = log4 (25)
x = log4 (25) + 3
x = approx. 5.321928
Dont' forget that to find a weird log, change the bases.
log4 (25) = ln(25) / ln(4)

[4]
log25 (750) = ln(750) / ln(25)
= approx. 2.05664137628
To 2 decimal places, it would be 2.06

Answer 2

1) First use the property
r log(x) = log(x^r)
4 log x = log(x^4)
2 log(x+3) = log[(x+3)^2]
Now use the property
log(a) + log(b) = log(ab)
log(x^4) + log[(x+3)^2]
= logV10 [(x^4)(x+3)^2]
2) logV7 (x+3) - 2 = logV7 (3)
logV7 (x+3) - logV7 (3) = 2
logV7 [(x+3)/3] = 2
(x+3)/3 = 7^2
(x+3)/3 = 49
x + 3 = 147
x = 144
3) Take the log of each side.
log(4^(x-3)) = log(25)
(x-3)log(4) = log(25)
x log(4) - 3 log(4) = log(25)
x log(4) = log(25) + 3 log(4)
x = (log(25) + 3 log(4))/log(4)
This is the exact answer. Feel free to use your calculator to find the approximate value.
4) logV25 750
= ln(750)/ln(25)
= 6.62007320653 / 3.21887582487
= 2.06
Note you could also use log (instead of natural log)
logV25 750 = log(750)/log(25) = 2.06

Answer 3

If i understand you correctly.

1)
4 logV10^x + 2 logV10^(x+3)
4log(10)x + 2log(10)(x + 3)
log(10)(x^4) + log(10)((x + 3)^2)
log(10)(x^4(x + 3)^2)
log(10)(x^4((x + 3)(x + 3)))
log(10)(x^4(x^2 + 3x + 3x + 9))
log(10)(x^4(x^2 + 6x + 9))
log(10)(x^6 + 6x^5 + 9x^4)
to take this further. log(10) = 1

ANS : log(x^6 + 6x^5 + 9x^4)

------------------------------------------------

2) Solve for x

logV7^(x+3) - 2 = logV7^3
log(7)(x + 3) - 2 = log(7)3
log(7)(x + 3) + log(7)3 = 2
log(7)(3(x + 3)) = 2
log(7)(3x + 9) = 2
This can be converted like this
7^2 = 3x + 9
49 = 3x + 9
40 = 3x
x = (40/3)
ANS : x = 13.33

--------------------------------------

3) Solve for x

4^x-3 = 25
4^(x - 3) = 25
(x - 3)ln4 = ln25
xln4 - 3ln4 = ln25
xln4 = ln25 + 3ln4
x = (ln25 + 3ln4)/(ln4)
(ln25 + ln64)/(ln4)
(2ln5 + 2ln8)/(2ln2)
(2(ln5 + ln8))/(2(ln2))
ANS : (ln5 + ln8)/(ln2)

If you don't believe me, plug in (ln25 + 3ln4)/(ln4) then divide that by (ln5 + ln8)/(ln2), i can guarantee you that they will divide into each other evenly.

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4) Approx to 2 decimal places

logV25^750
log(25)750
(log(750))/(log(25))
(log(25 * 30))/(log(25))
(log(25) + log(30))/(log(25))
1 + (log(30)/log(25))
1 + log(25)30
ANS : 2.06

Answer 4

I can see why you failed.... lol. Good luck. Hope someone can help you out.