2log3 5+log3 30+3log3 6
Good
2 log3 ( 5 ) + log3 ( 30 ) + 3 log3 ( 6 ) =
log3 ( 5 ^ 2 ) + log3 ( 30 ) + log3 ( 6 ^ 3 ) =
log3 ( 25 ) + log3 ( 30 ) + log3 ( 216 ) =
log3 ( 25 ∙ 30 ∙ 216 ) =
log3 ( 162 000 ) = 10.91863957643961125
This mean:
3 ^ 10.91863957643961125 = 162 000
To simplify the expression 2log3 5 + log3 30 + 3log3 6, we will use logarithmic properties.
1. Start with the expression: 2log3 5 + log3 30 + 3log3 6.
2. Rewrite the exponents using the logarithmic definition: log3 5^2 + log3 30 + log3 6^3.
3. Apply the power rule for logs: 2log3 5 + log3 30 + log3 216.
4. Calculate the logarithmic terms: log3 25 + log3 30 + log3 216.
5. Combine the logarithmic terms into a single logarithm: log3 (25 * 30 * 216).
6. Simplify the expression inside the logarithm: log3 (162,000).
7. Evaluate the logarithm: log3 (162,000) ≈ 6.2788.
Therefore, 2log3 5 + log3 30 + 3log3 6 simplifies to approximately 6.2788.
To simplify the expression 2log3 5 + log3 30 + 3log3 6:
Step 1: Apply the properties of logarithms. Use the logarithmic identity log(base a) b + log(base a) c = log(base a) (b * c).
- In this case, apply this property to combine the first two terms: 2log3 5 + log3 30 = log3 (5^2 * 30).
- Simplify further: 2log3 5 + log3 30 = log3 (25 * 30) = log3 750.
Step 2: Simplify the expression further. Now the expression becomes log3 750 + 3log3 6.
- Again, apply the logarithmic identity log(base a) b + log(base a) c = log(base a) (b * c), but this time for the last two terms: 3log3 6 = log3 (6^3).
- Simplify further: 3log3 6 = log3 (216).
Step 3: Combine the logarithms. Now the expression becomes log3 750 + log3 (216).
- Apply another logarithmic identity: log(base a) b + log(base a) c = log(base a) (b * c).
- Combine the terms: log3 750 + log3 (216) = log3 (750 * 216).
Step 4: Simplify the expression further. Now the expression becomes log3 (750 * 216).
- Multiply the numbers within the logarithm: log3 (750 * 216) = log3 162,000.
The final simplified expression is log3 162,000.