If log10^9=0.95 and log10^2=0.30, find the following:

1. log10^9/2
2. log10^3
3. log10^36
4. log10^20/9
5. log10^900
6. log10^1/9
7. log10^1/2000
8. log10^162

I need to see all of the steps.
Thanks

Sure! I'll provide step-by-step solutions for each of the questions:

1. log10^9/2: We can use the property of logarithms that states log_a(b/c) = log_a(b) - log_a(c). Applying this property, we have log10^9/2 = log10^9 - log10^2. Given that log10^9 = 0.95 and log10^2 = 0.30, we substitute these values into the equation: log10^9/2 = 0.95 - 0.30 = 0.65.

2. log10^3: Here, we have log10^3. Using the fact that log_a(b^c) = c * log_a(b), we can rewrite the expression: log10^3 = 3 * log10. Since log10 = 1, we can find the solution: log10^3 = 3 * 1 = 3.

3. log10^36: Similar to question 2, we can rewrite log10^36 as 36 * log10. Given that log10 = 1, we can solve: log10^36 = 36 * 1 = 36.

4. log10^20/9: Using the property mentioned in question 1, we can write log10^20/9 as log10^20 - log10^9. Substituting the given values, we get log10^20/9 = 0.95 - 0.30 = 0.65.

5. log10^900: As mentioned earlier, log_a(b^c) = c * log_a(b). So log10^900 can be rewritten as 900 * log10. Since log10 = 1, the solution is log10^900 = 900 * 1 = 900.

6. log10^1/9: To simplify this expression, we can use the fact that log_a(b^c) = c * log_a(b). Rewriting log10^1/9 as (1/9) * log10, the final step is to evaluate it: log10^1/9 = (1/9) * 1 = 1/9.

7. log10^1/2000: By the same logic as in question 6, we can rewrite log10^1/2000 as (1/2000) * log10. Since log10 = 1, we can find the solution: log10^1/2000 = (1/2000) * 1 = 1/2000.

8. log10^162: Expanding on the rule log_a(b^c) = c * log_a(b), we can rewrite log10^162 as 162 * log10. Since log10 = 1, the solution is log10^162 = 162 * 1 = 162.

I hope this helps! Let me know if you have any further questions.

Sure! Let's solve each of these step by step:

1. To find log10^9/2, we can use the property of logarithms that states log(a/b) = log(a) - log(b). Applying this, we have:
log10^9/2 = log10^9 - log10^2.
Using the given values, we get:
log10^9/2 = 0.95 - 0.30 = 0.65.

2. To find log10^3, we can use the property that log(a^b) = b * log(a). Applying this to our case, we have:
log10^3 = 3 * log10^1.
Since log10^1 is equal to 0 (as it is the base of the logarithm), we get:
log10^3 = 3 * 0 = 0.

3. To find log10^36, we can apply the property log(a^b) = b * log(a) once again:
log10^36 = 36 * log10^1.
Since log10^1 is equal to 0, we get:
log10^36 = 36 * 0 = 0.

4. To find log10^20/9, we can use the property log(a/b) = log(a) - log(b):
log10^20/9 = log10^20 - log10^9.
Using the given values, we have:
log10^20/9 = 0.95 - 0.90 = 0.05.

5. To find log10^900, we can use the property log(a^b) = b * log(a):
log10^900 = 900 * log10^1.
Since log10^1 is equal to 0, we get:
log10^900 = 900 * 0 = 0.

6. To find log10^1/9, we can apply the property log(a^(-b)) = -b * log(a):
log10^1/9 = -1/9 * log10^9.
Using the given value, we get:
log10^1/9 = -1/9 * 0.95 = -0.1056.

7. To find log10^1/2000, we again apply the property log(a^(-b)) = -b * log(a):
log10^1/2000 = -1/2000 * log10^2000.
Using the given value, we have:
log10^1/2000 = -1/2000 * 3.30 = -0.00165.

8. Finally, to find log10^162, we directly use the given value:
log10^162 = 0.95.

Hope that helps!