log2+3log2-4log7
since sum(logs) = log(product) you have
log(2*2^3/7^4) = log 16/2401
To simplify the expression log2 + 3log2 - 4log7, we can use logarithmic rules:
1. The rule log a + log b = log (a * b) states that the sum of logarithms of two numbers is equal to the logarithm of their product.
2. The rule log a - log b = log (a / b) states that the difference of logarithms of two numbers is equal to the logarithm of their quotient.
Using these rules, we can simplify the expression step by step:
Step 1: Combine the two log2 terms using the rule log a + log b = log (a * b):
log2 + 3log2 = log2 + log2^3 = log2 * 2^3 = log2 * 8 = log 16 = 4
Step 2: Simplify the log7 term:
4log7 = log7^4 = log 2401
Therefore, the simplified expression is log 2401.
To simplify the expression log2 + 3log2 - 4log7, we can use logarithmic properties. Let's break it down step by step:
1. Start with log2 + 3log2 - 4log7.
2. Combine the logarithms with the same base by using the logarithmic property:
log a + log b = log (a * b).
Applying this property to our expression, we get:
log2 + log2^3 - log7^4.
3. Simplify the exponents:
log2 + log8 - log2401.
4. Use another logarithmic property:
log a^b = b * log a.
Equation becomes:
log2 + 3log2 - 4log7.
log2 + log2^3 - log7^4.
log2 + log8 - log2401.
log2 + 3log2 - 4log7.
5. Simplify further:
log2 + log2^3 - log7^4 = log2 + log8 - log2401
log2 + 3log2 - 4log7.
6. Evaluate the logarithmic expressions:
log2 = 1 (since 2^1 = 2)
log8 = 3 (since 2^3 = 8)
log7 = a numerical value.
Therefore, the expression is simplified to:
1 + 3 - 4log7.
Note: To find the numerical value of log7, you can use a calculator or a logarithm table.
log ( 2 * 2 ^ 3 / 7 ^ 4 ) =
log ( 2 ^ 4 / 7 ^ 4 ) =
log [ ( 2 / 7 ) ^ 4 ] =
4 ∙ [ log ( 2 ) - log ( 7 ) ] =
4 ∙ ( 0.69314718056 - 1.945910149 ) =
4 ∙ ( - 1.25276296845 ) =
- 5.01105187398
OR
log ( 2 * 2 ^ 3 / 7 ^ 4 ) =
log ( 2 * 8 / 2401 ) =
log ( 16 / 2401 ) =
log ( 16 ) - log ( 2401 ) =
2.77258872224 - 7.78364059622 =
- 5.01105187398
In this case:
log is the natural logarithm