Log384÷5+log81÷32+3log5÷3-log9=2
I assume you mean
log(384/5)+log(81/32)+3log(5/3)-log9
=log(384*81*125)/(5*32*27*9)
=log(3888000/38880)
=log 100
= 2
To solve the equation log384÷5+log81÷32+3log5÷3-log9=2, we will simplify the expression step by step.
Step 1: Apply the rules of logarithms to solve the equation. We can use the following log rules:
- log(a/b) = log(a) - log(b)
- log(a^n) = n * log(a)
- log(a * b) = log(a) + log(b)
Given equation: log384÷5+log81÷32+3log5÷3-log9=2
Step 2: Simplify the terms using the log rules.
Using log(a/b) = log(a) - log(b):
log(384) - log(5) + log(81) - log(32) + 3(log(5) - log(3)) - log(9) = 2
Step 3: Apply the logarithm rules further to simplify the equation.
Using properties of exponents and logarithms:
log(384) - log(5) + log(81) - log(32) + log(5^3) - log(3^3) - log(9) = 2
Step 4: Simplify the terms further.
Using log(a^n) = n * log(a):
log(384) - log(5) + log(81) - log(32) + 3log(5) - 3log(3) - log(9) = 2
Step 5: Combine the terms.
log(384) + log(81) + 3log(5) - log(5) - log(32) - 3log(3) - log(9) = 2
Step 6: Use properties of logarithms to combine the terms.
log(384 * 81) + log(5^3) - log(32) - log(3^3) - log(9) = 2
log(31104) + log(125) - log(32) - log(27) - log(9) = 2
Step 7: Combine the terms using properties of logarithms.
log(31104*125) - log(32) - log(27) - log(9) = 2
log(3888000) - log(32) - log(27) - log(9) = 2
Step 8: Apply the logarithm rules to further simplify.
log(3888000) - log(32*27*9) = 2
log(3888000) - log(7776) = 2
Step 9: Combine the terms.
log(3888000/7776) = 2
log(500) = 2
Step 10: Find the value of the logarithm.
The logarithm with the base 10 of 500 is approximately 2.698.
Therefore, the equation log384÷5+log81÷32+3log5÷3-log9=2 is not true because log(384÷5)+log(81÷32)+3log(5÷3)-log(9) does not equal 2.
To solve the equation Log384÷5+log81÷32+3log5÷3-log9=2, we need to apply several logarithm rules and simplify the expression step by step.
Let's break it down:
1. Start by applying the division rule of logarithms: Log(a/b) = Log(a) - Log(b).
Log384÷5 becomes Log384 - Log5.
2. Next, simplify Log384 by using logarithm rules. Since there is no base mentioned, we'll assume a base of 10.
Log384 = Log10(384).
We can rewrite 384 as a multiplication of factors of 10: 384 = 3 * 10^2 + 8 * 10^1 + 4 * 10^0.
Applying the logarithm rule Log(a^b) = b * Log(a), we can write Log10(384) as:
Log10(3 * 10^2 + 8 * 10^1 + 4 * 10^0) = Log10(3) + Log10(10^2) + Log10(8) + Log10(10^1) + Log10(4) + Log10(10^0).
Simplifying, we have Log10(3) + 2 + Log10(8) + 1 + Log10(4) + 0.
Combining like terms, we get Log10(3) + Log10(8) + Log10(4) + 3.
3. Continuing from step 1 and 2, we have:
Log384 - Log5 + Log81÷32 + 3log5÷3 - log9 = 2.
Simplifying further, we have:
(Log10(3) + Log10(8) + Log10(4) + 3) - Log5 + log81÷32 + 3log5÷3 - log9 = 2.
4. Apply the division rule of logarithms to Log81÷32 and 3log5÷3.
Log81÷32 becomes Log81 - Log32.
3log5÷3 becomes 3 * (Log5 - Log3).
5. Substitute the expressions:
(Log10(3) + Log10(8) + Log10(4) + 3) - Log5 + (Log81 - Log32) + 3 * (Log5 - Log3) - log9 = 2.
6. Next, simplify each term:
(Log10(3) + Log10(8) + Log10(4) + 3) - Log5 + (Log81 - Log32) + 3 * (Log5 - Log3) - log9 = 2.
Combine like terms:
Log10(3 * 8 * 4) + 3 - Log5 + Log81 - Log32 + 3 * Log5 - 3 * Log3 - Log9 = 2.
Log10(3*8*4) = Log10(96).
log(96) = log(2^5 * 3) = 5 * log(2) + log(3) = 5(0.3010) + 0.4771 ≈ 1.5566.
The equation becomes:
1.5566 + 3 - Log5 + Log81 - Log32 + 3* Log5 - 3* Log3 - Log9 = 2.
7. Finally, combine like terms:
4.5566 - Log5 + Log81 - Log32 + 3* Log5 - 3* Log3 - Log9 = 2.
Rearranging the terms:
- Log5 + Log81 - Log32 + 3* Log5 - 3* Log3 - Log9 = 2 - 4.5566.
Compute the right-hand side:
2 - 4.5566 = -2.5566.
The equation becomes:
- Log5 + Log81 - Log32 + 3* Log5 - 3* Log3 - Log9 = -2.5566.
8. Solve for the logarithmic terms:
- Log5 + Log81 - Log32 + 3* Log5 - 3* Log3 - Log9 = -2.5566.
Using logarithmic properties and rules, we can simplify this further. However, it seems that there may have been an error during the calculations, as the logarithmic terms cannot be simplified to obtain an equation equal to -2.5566.
Please double-check the calculations and the original equation Log384÷5+log81÷32+3log5÷3-log9=2 and ensure the equation is correctly represented.