Evaluation without using a calculator or mathematical tables :log 32+log 128-log 729/log 32+log 2-log 27
To compute this expression without a calculator or mathematical tables, we can simplify and manipulate the expression to work with more manageable numbers.
First, let's focus on the numerator:
log 32 + log 128
We can use logarithmic properties to simplify this:
log 32 + log 128 = log (32 * 128) = log (4096)
Now, looking at the denominator:
log 729/log 32 + log 2 - log 27
We can again use logarithmic properties to simplify this:
log 729/log 32 + log 2 - log 27 = log (729/32) + log 2 - log 27
Next, let's work on simplifying 729/32:
729/32 = 9 * 81 / (4 * 8) = (9/2) * (81/8) = (9/2) * (9/1) = 81/2
Now we can rewrite the expression as:
log (81/2) + log 2 - log 27
Let's simplify further:
log (81/2) + log 2 - log 27 = log ((81/2) * 2 / 27) = log (162 / 27) = log 6
Therefore, the expression simplifies to log 6.
To evaluate the expression without using a calculator or mathematical tables, we can simplify each logarithmic term separately using the following logarithmic properties:
1. log(ab) = log(a) + log(b)
2. log(a/b) = log(a) - log(b)
Let's simplify each term step-by-step:
1. log 32 + log 128 - log 729
Using property 1, we can combine the first two terms:
log(32 * 128) - log(729)
Taking the logarithm base 10 of the numbers inside:
log(4096) - log(729)
Both 4096 and 729 can be expressed as powers of 10:
log(10^3) - log(10^2.86)
Simplifying the expression further:
3 - 2.86 = 0.14
2. log 32 + log 2 - log 27
Following the same steps as before, we combine the logarithmic terms:
log(32 * 2) - log(27)
Taking the logarithm base 10 of the numbers inside:
log(64) - log(27)
Again, expressing the numbers as powers of 10:
log(10^1.81) - log(10^1.43)
Further simplification:
1.81 - 1.43 = 0.38
Finally, we substitute the simplified values back into the original expression:
0.14 + 0.38 = 0.52
Therefore, the value of the given expression is 0.52.