6^log of base 6 is 5 +3^log of base 9 is 16 ????? pls give me solution...
Solution to 6^log of base 6 is 5 and 3^log of base 9 is 16
Finding the Value of Logarithm
To solve this problem, we need to first understand the concept of logarithm. A logarithm is the power to which a number must be raised to get another number. In other words, it tells us what exponent we need to use to get a certain number.
For example, if we have the equation:
2^x = 8
We can rewrite this as:
x = log2 (8)
Here, log2 (8) means the logarithm of 8 with base 2. In other words, it tells us what power of 2 we need to use to get 8. In this case, it is 3, since 2^3 = 8.
Now, let's apply this concept to the given equations:
6^log6 (x) = 5
3^log9 (y) = 16
Here, log6 (x) means the logarithm of x with base 6. Similarly, log9 (y) means the logarithm of y with base 9.
Our goal is to find the value of x and y that satisfy these equations. To do this, we need to first find the values of log6 (x) and log9 (y) using the given information.
Using the Laws of Logarithm
To find the values of log6 (x) and log9 (y), we can use the laws of logarithm. These laws tell us how to simplify logarithmic expressions.
In particular, we can use the following laws:
- logb (x * y) = logb (x) + logb (y)
- logb (x / y) = logb (x) - logb (y)
- logb (x^y) = y * logb (x)
Using these laws, we can simplify the given equations as follows:
6^log6 (x) = 5
log6 (x) * log6 (6) = log6 (5)
log6 (x) = log6 (5) / log6 (6)
log9 (y) * log3 (3) = log9 (16)
log9 (y) * 1 = log9 (16) / log9 (3)
log9 (y) = log9 (16) / log9 (3)
Calculating the Value of Logarithm
Now that we have simplified the equations, we can calculate the values of log6 (x) and log9 (y) using a calculator or by hand using the change of base formula.
Using a