Log 384/5 + log 81/32 + 3log 5/3 + log 1/9
Log 384/5 + log 81/32 + 3log 5/3 + log 1/9
by the rules of logs we get
= log 384 - log 5 + log 81 - log 32 + 3log 5 - 3log 3 + log 1 - log 9
= log 3 + log 128 - log 5 + 4log 3 - 5log 2 + 3log 5 - 3log 3 + 0 - 2 log3
= 0log3 + 7log2 - log5 - 5log2 + 3log5
= 2log2 + 2log5
= log4 + log25 = log(4*25)
= log 100
= 2
or, easier way
Log 384/5 + log 81/32 + 3log 5/3 + log 1/9
= log(384/5*81/32*125/27*1/9)
= log 100
= 2
factor 384, 81,32,9
you should be able to factor into 3,5,and 2 factors. Example:
384=3*128=3*4*32=3*4*4*4*2=3* 2^7
so the log of 384=log3+7log2 so the first term Log 384/5 becomes
7log2+log3-log5
do that same thing with the other terms.
Oh boy, we have a math problem on our hands! Let's see if we can solve it together.
Do you know what a log is? It's like a lumberjack's favorite button on the calculator. Just kidding!
But seriously, a logarithm is the inverse operation of exponentiation. It tells us what exponent we need to raise a particular base to in order to get a given value.
In this case, we have a bunch of logarithmic expressions that we need to add together. Let's simplify each one:
log 384/5 = log(384) - log(5)
log 81/32 = log(81) - log(32)
3log 5/3 = log((5/3)^3) = log(125/27) = log(125) - log(27)
log 1/9 = log(1) - log(9)
Now, let's plug these simplified expressions back into the original equation and combine like terms:
(log(384) - log(5)) + (log(81) - log(32)) + (log(125) - log(27)) + (log(1) - log(9))
Now, let's use the logarithmic property that states log(a) - log(b) = log(a/b):
log(384/5) + log(81/32) + log(125/27) + log(1/9)
Now, we can combine everything into a single logarithmic expression:
log((384/5) * (81/32) * (125/27) * (1/9))
And we can simplify the expression inside the logarithm:
log((92160/40) * (405/32))
log(2304 * 405/32)
log(737280/32)
log(23040)
So, the final answer is log(23040). And the answer is...
drumroll, please...
log(23040)!
Remember, math can be a bit tricky, but with a little humor, we can make it more bearable.
To simplify the expression log 384/5 + log 81/32 + 3log 5/3 + log 1/9, we will start by applying logarithmic rules.
1. Recall the rule log a + log b = log (a * b). Applying this rule to the first two terms, we have:
log 384/5 + log 81/32 = log (384/5 * 81/32)
2. Next, simplify the multiplication inside the logarithm:
384/5 * 81/32 = 124416/160 = 7776/10 = 777.6
3. The expression now becomes log 777.6 + 3log 5/3 + log 1/9.
4. Recall the rule log a^b = b * log a. Applying this rule, we have:
3log 5/3 = log (5/3)^3 = log(125/27)
5. The expression now becomes log 777.6 + log (125/27) + log 1/9.
6. Recall the rule log a + log b = log (a * b). Applying this rule to the last two terms, we have:
log (125/27) + log 1/9 = log ((125/27) * 1/9) = log (125/243)
7. The expression now becomes log 777.6 + log (125/243).
8. Recall the rule log a + log b = log (a * b). Applying this rule to the final two terms, we have:
log 777.6 + log (125/243) = log (777.6 * (125/243))
9. Simplify the multiplication inside the logarithm:
777.6 * (125/243) ≈ 427.36
10. The final expression is log 427.36.
To simplify this expression, we can use the logarithmic properties and combine the logarithms:
1. log 384/5 + log 81/32 + 3log 5/3 + log 1/9
Using the logarithmic property log a + log b = log (a * b), we can rewrite this expression as:
2. log[(384/5) * (81/32) * (5/3)^3 * (1/9)]
Next, we simplify the values within the parentheses:
3. log[(76/1) * (81/32) * (125/27) * (1/9)]
Now, we combine these fractions together:
4. log[(76 * 81 * 125) / (1 * 32 * 27 * 9)]
Simplifying further:
5. log(1233000 / 69984)
Now we'll find the decimal value of this expression using a calculator or logarithmic tables:
6. log(1233000 / 69984) ≈ log(17.6186) ≈ 1.2458
Therefore, the simplified value of the expression log 384/5 + log 81/32 + 3log 5/3 + log 1/9 is approximately 1.2458.