Given that log3=0.4771 and log5=0.6690 and log2=0.3010. Without using tables or calculators evaluate log0.243
.243 = 243 / 100 = (3 * 81) / (10 * 10) = 3^5 / (2^2 * 5^2)
log(0.243) = 5log(3) - 2[log(2) + log(5)]
FYI ... log(5) = 0.6990
answer
To evaluate log0.243 without using tables or calculators, we can use the properties of logarithms.
1. Start by rewriting 0.243 as a fraction with a power of 10. Since 10 is not a factor of 243, we can choose a power of 10 that will make the calculation easier. Let's rewrite 0.243 as 243/1000.
2. Rewrite the logarithm equation using the properties of logarithms:
log0.243 = log(243/1000)
3. Use the logarithmic property that states: log(a/b) = log(a) - log(b). Apply this property to our equation:
log0.243 = log(243) - log(1000)
4. Use the logarithmic properties:
log(243) = log(3^5)
log(1000) = log(10^3)
We can use the property log(a^b) = b * log(a) to simplify further:
log0.243 = 5 * log(3) - 3 * log(10)
5. We are given log(3), log(10), and log(2). Substitute these values into the equation:
log0.243 = 5 * 0.4771 - 3 * 0.3010
6. Perform the calculations:
log0.243 = 2.3855 - 0.9030
7. Simplify the result:
log0.243 ≈ 1.4825
To evaluate log0.243 without using tables or calculators, we need to use logarithmic properties.
First, let's rewrite 0.243 as a power of 10:
0.243 = 2.43 × 0.1 = (3 × 0.81) × 0.1 = 3 × 0.081
Now, we can use the logarithmic property that states log(x * y) = log(x) + log(y):
log0.243 = log(3 * 0.081)
Next, we can use another logarithmic property that states log(x^n) = n * log(x):
log(3 * 0.081) = log(3) + log(0.081) = log(3) + log(3^2) - log(10^2)
Now, using log properties, we can simplify further:
log(3) + log(3^2) - log(10^2) = log(3) + 2log(3) - 2log(10)
Since log(10) = 1, we can simplify further:
log(3) + 2log(3) - 2log(10) = log(3) + 2log(3) - 2
Now, substitute the given values:
log(3) = 0.4771
Thus, we can evaluate:
log0.243 = 0.4771 + 2 * 0.4771 - 2 = 0.4771 + 0.9542 - 2 = 1.431 - 2 = -0.569
Therefore, log0.243 ≈ -0.569.