evaluate without using a calculator or mathematical tables:log32+log128-log729/log32+log2+log27
To evaluate this expression without using a calculator or mathematical tables, we will use the basic properties of logarithms:
1. log(a*b) = log(a) + log(b)
2. log(a/b) = log(a) - log(b)
Let's break down the expression step by step:
1. log32 + log128 - log729 = log(32*128/729)
Using property 1, we can simplify this to:
log(4096/729)
2. log32 + log2 + log27 = log(32*2*27)
Using property 1 again, we simplify this to:
log(3456)
Now, our expression becomes:
log(4096/729) / log(3456)
To evaluate this further, we need to find the common logarithm of both the numerator and the denominator. Let's now try to simplify each individually:
1. Numerator: log(4096/729)
We need to find numbers whose quotient is equal to 4096/729. Let's list some powers of 2 and 3 to identify a pair:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
...
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
From this listing, we can see that 4096 is equal to 2^12 (2 raised to the power of 12) and 729 is equal to 3^6. Therefore,
4096/729 = (2^12)/(3^6) = (2/3)^6
Now, the numerator becomes:
log((2/3)^6)
Using property 2, we can simplify this to:
6 * log(2/3)
2. Denominator: log(3456)
We can use the same strategy here to find numbers whose product is equal to 3456. Let's list some factors of 3456:
2 * 2 = 4
3 * 2 = 6
3 * 2 * 2 = 12
4 * 4 = 16
16 * 4 = 64
64 * 3 = 192
192 * 2 = 384
384 * 2 = 768
768 * 4 = 3072
3072 * 2 = 6144
3072 * 2 * 2 = 12288
12288 * 2 = 24576
24576 * 2 = 49152
49152 * 2 = 98304
98304 * 3 = 294912
294912 * 2 = 589824
589824 * 2 = 1179648
1179648 * 2 = 2359296
2359296 * 2 = 4718592
From this listing, we can see that 3456 = 2^5 * 3^3 * 4. Therefore,
3456 = 2^5 * 3^3 * 4 = (2 * 3 * 2)^3 * 4 = (12)^3 * 4
Now, the denominator becomes:
log((12)^3 * 4)
Using property 1, we can simplify this to:
3 * log(12) + log(4)
Now, our expression becomes:
(6 * log(2/3)) / (3 * log(12) + log(4))
At this stage, we cannot simplify it any further without a calculator or mathematical tables.
To evaluate the expression without using a calculator or mathematical tables, we can use the properties of logarithms.
Let's simplify each part step by step:
1. log32 + log128
First, let's simplify by using the logarithmic property for addition:
log32 + log128 = log(32 * 128)
To simplify further, we can calculate the product of 32 and 128 mentally:
log32 + log128 = log(4096)
2. log729 / log32
Second, let's simplify the division using the logarithmic property:
log729 / log32 = log(729) - log(32)
We can evaluate each logarithm separately:
log(729) = 3 (since 10^3 = 1000 is the closest power of 10 to 729)
log(32) = 1.505 (approximately, since 2^1.505 is the closest power of 2 to 32)
3. log2 + log27
Lastly, let's simplify the addition using the logarithmic property:
log2 + log27 = log(2 * 27)
Evaluating the multiplication:
log2 + log27 = log(54) = 1.732 (approximately, since 10^1.732 is the closest power of 10 to 54)
Now, we have all the simplified values, so let's substitute them back into the original expression:
log32 + log128 - log729 / log32 + log2 + log27 = log(4096) - 3 / 1.505 + 1.732
At this point, the expression cannot be simplified further without using a calculator or tables.