Calculate without using tables or calculators log 450 and log 324 given log 2 =0.3010 and log 3 =0.4771

log 450= log9 × log50

log3^2 × log 100÷2
Log100=2 So it's 2log3 × log 100÷2
2 (0.4771) × 2 - 0.3010
0.9542 + 1.699
2.6532

To calculate log 450 and log 324 using the given values, we need to use the logarithmic properties. Specifically, we'll use the property that states log (a * b) = log(a) + log(b).

First, let's solve for log 450:
We know that log 450 = log (9 * 50).

Using the logarithmic property, we can rewrite this as:
log 450 = log 9 + log 50.

Since 9 can be written as 3^2, we have:
log 450 = log (3^2) + log 50.

Using the given value log 3 = 0.4771, we can substitute it in:
log 450 = 2 * log 3 + log 50.

Now, substitution for log 2 = 0.3010 and log 3 = 0.4771 gives:
log 450 = 2 * 0.4771 + log 50.

Calculating the sum gives us:
log 450 = 0.9542 + log 50.

Now, let's solve for log 50:
Using logarithmic property, log 50 = log (10 * 5).

Applying the property again, log 50 = log 10 + log 5.

Since log 10 = 1 (as 10 = 10^1), we have:
log 50 = 1 + log 5.

Now, let's solve for log 5:
Using logarithmic property, log 5 = log (10 / 2).

Applying the property again, log 5 = log 10 - log 2.

Since log 10 = 1, and log 2 = 0.3010, we have:
log 5 = 1 - 0.3010.

Simplifying the expression gives us:
log 5 = 0.6990.

Now, substituting this value back into the previous equation for log 50, we have:
log 50 = 1 + log 5 = 1 + 0.6990 = 1.6990.

Substituting the value of log 50 back into the equation for log 450, we have:
log 450 = 0.9542 + log 50 = 0.9542 + 1.6990 = 2.6532.

Therefore, log 450 is approximately 2.6532.

Now, let's move on to calculate log 324:
We know that log 324 = log (9 * 36).

Using the logarithmic property, we can rewrite this as:
log 324 = log 9 + log 36.

Since 9 can be written as 3^2, and 36 can be written as 6^2, we have:
log 324 = log (3^2) + log (6^2).

Using the given values log 3 = 0.4771 and log 2 = 0.3010, we can substitute them in:
log 324 = 2 * log 3 + 2 * log 6.

Now, let's solve for log 6:
Using logarithmic property, log 6 = log (2 * 3).

Applying the property again, log 6 = log 2 + log 3.

Since log 2 = 0.3010 and log 3 = 0.4771, we have:
log 6 = 0.3010 + 0.4771.

Simplifying the expression gives us:
log 6 = 0.7781.

Now, substituting this value back into the equation for log 324, we have:
log 324 = 2 * log 3 + 2 * log 6 = 2 * 0.4771 + 2 * 0.7781 = 0.9542 + 1.5562 = 2.5104.

Therefore, log 324 is approximately 2.5104.

To calculate log 450 and log 324 without using tables or calculators, we can utilize the properties of logarithms and the given values of log 2 and log 3. Here's how:

1. Start by expressing 450 and 324 as powers of 2 and 3:
- 450 = 2^x
- 324 = 3^y

2. To find the exponent x, let's take the logarithm base 2 of both sides of the first equation:
log 450 = log (2^x)
Applying the properties of logarithms (log a^b = b * log a), we get:
log 450 = x * log 2

3. We can substitute the value of log 2 (0.3010) into the equation:
log 450 = x * 0.3010

4. Now, solve for x by rearranging the equation:
x = log 450 / 0.3010

5. Calculate the value of log 450 by dividing log 450 by log 2:
log 450 = log 450 / 0.3010

6. Repeat the same process to find the value of log 324:
- Express 324 as a power of 3: 324 = 3^y
- Take the logarithm base 3 of both sides: log 324 = log (3^y)
- Apply the property of logarithms: log 324 = y * log 3
- Substitute the value of log 3 (0.4771) into the equation: log 324 = y * 0.4771
- Solve for y: y = log 324 / 0.4771
- Calculate the value of log 324: log 324 = log 324 / 0.4771

Remember to perform the divisions correctly to obtain the final results for log 450 and log 324.

I will do the second one

324 = 4(81)
= 2^2 x 3^4

log324 = log(2^2 x 3^4)
= log 2^2 + log 3^4
= 2log2 + 4log3
= 2(.3010) + 4(.4771)
= .6020 + 1.9084
= 2.5104

log(450) = log(10x45)
= log(10x9x5)
= log10 + log 3^2 + log5 ---> need log5

other combinations:
450 = 5^2 x 9x2 = 5^2 x 3^2 x 2 --- still need log 5