3log2 +log20 -log1.6
Using logarithm rules, we can simplify this expression as follows:
3log2 + log20 - log1.6
= log(2^3) + log(20) - log(1.6) (logarithm rule: log(a) + log(b) = log(ab))
= log(8) + log(20) - log(1.6) (2^3 = 8)
Now, let's use another logarithm rule: log(a) - log(b) = log(a/b)
= log(8 * 20 / 1.6) (logarithm rule: log(a) + log(b) = log(ab))
= log(160)
Therefore, the simplified expression is log(160).
To simplify the expression 3log2 + log20 - log1.6, we can use logarithm rules to combine the terms:
1. Start with the given expression: 3log2 + log20 - log1.6
2. Use the power rule of logarithms, which states that log_a(b^c) = c * log_a(b). Apply this rule to simplify each term separately:
- For the term 3log2, use the power rule to get: 3 * log2(2) = 3 * 1 = 3
- For the term log20, it is not a logarithm of a power, so we leave it as it is.
- For the term log1.6, it is also not a logarithm of a power, so we leave it as it is.
Therefore, the simplified expression becomes: 3 + log20 - log1.6
3. Solve the logarithms:
- Go to the first term, log20. We can rewrite this as log2(10) since log_a(b) = log_c(b) / log_c(a) using the change of base formula, where c is any positive number such that c≠1.
log2(10) is approximately 3.3219 (rounded to 4 decimal places).
- Now, move to the second term, log1.6. This can be written as log2(1.6) using the change of base formula.
log2(1.6) is approximately 0.7370 (rounded to 4 decimal places).
4. Substitute the values obtained from the step above back into the original expression:
Simplified expression = 3 + log20 - log1.6
= 3 + 3.3219 - 0.7370
5. Perform the addition and subtraction operations:
Simplified expression ≈ 2.5849 (rounded to 4 decimal places).
Therefore, the simplified expression is approximately 2.5849 (rounded to 4 decimal places).
To solve this expression, you can follow these steps:
Step 1: Use the logarithmic rules to simplify the expression.
- The rule "log(a) + log(b) = log(ab)" states that the sum of logarithms with the same base is equal to the logarithm of the product of their arguments.
- The rule "log(a) - log(b) = log(a/b)" states that the difference of logarithms with the same base is equal to the logarithm of the division of their arguments.
Applying these rules to the expression:
3log2 + log20 - log1.6 = log(2^3) + log(20) - log(1.6)
Step 2: Simplify the logarithmic expression by evaluating the logarithms.
- log(2^3) = log(8) since 2^3 = 8
- log(20) = log(10 * 2) = log(10) + log(2) since 20 can be expressed as the product of 10 and 2, and we can use the logarithmic rule mentioned earlier.
- log(1.6) = log(10/6.25) = log(10) - log(6.25) since 1.6 can be expressed as the division of 10 and 6.25, and we can use the logarithmic rule mentioned earlier.
Step 3: Evaluate the simplified expression.
Substituting the simplified logarithms back into the expression:
log(8) + log(10) + log(2) - log(10) + log(6.25) = log(8) + log(2) + log(6.25)
Step 4: Calculate the value of the logarithms.
- log(8) is the power you have to raise the base 10 to get 8. In this case, log(8) ≈ 0.9031.
- log(2) is the power you have to raise the base 10 to get 2. In this case, log(2) ≈ 0.3010.
- log(6.25) is the power you have to raise the base 10 to get 6.25. In this case, log(6.25) ≈ 0.7959.
Step 5: Add up the logarithmic values.
log(8) + log(2) + log(6.25) ≈ 0.9031 + 0.3010 + 0.7959 ≈ 1.999
So, the simplified expression 3log2 + log20 - log1.6 is approximately equal to 1.999.