I need help with these logs
1: Log underscore4 16X
2: 2 Ln ( X(sqr. root)e
3:5^[2logUnderscore5(3x)]
4:log underscore2 ^[1280-logunderscore2 5)
I don't understand how to do them thanks
To solve these logarithmic expressions, let's break them down step by step. I'll explain each part so that you can understand and solve similar problems in the future.
1: Log underscore4 16X
This logarithmic expression is written in base-4 logarithm form. To solve it, we need to rewrite it in exponential form. Recall that the base of a logarithm is the number being raised to a power.
Exponential form: 4^y = x
So, we need to find the value of 'y'. In this case, x is given as 16X. We can write the equation as follows:
4^y = 16X
Now, we need to find the exponent 'y' that makes the equation true. To do that, we can rewrite 16X as 4^2X (since 16 = 4^2), which gives us:
4^y = 4^2X
To solve this equation, set the exponents equal to each other:
y = 2X
So, the solution to the logarithmic expression is y = 2X.
2: 2 Ln ( X(sqr. root)e
In this expression, we have the natural logarithm (Ln) of a product that includes the square root of X and the constant 'e.' To simplify this expression, we'll use some logarithmic properties.
First, recall that Ln is the logarithm with base 'e.' So, Ln(e) = 1.
Using this property, we can rewrite the expression as follows:
2(1) = 2
So, the solution to this logarithmic expression is 2.
3: 5^[2logUnderscore5(3x)]
This expression involves logarithmic functions and exponentiation. To solve it, we'll apply logarithmic properties.
Notice that we have a logarithm with base 5 inside the exponent. We can use the property that states: log base a (b^c) = c * log base a (b).
Using this property, we can rewrite the expression as follows:
5^[2 * log base 5 (3x)]
Now, applying the logarithmic property, we have:
5^[log base 5 ((3x)^2)]
Again, using another logarithmic property, we can simplify to:
5^[(3x)^2]
So, the solution to this expression is 5^[(3x)^2].
4: log underscore2 ^[1280-logunderscore2 5)
In this expression, we have a logarithmic function with base 2 raised to the power of another logarithmic function. Let's apply logarithmic properties to simplify it.
Using the property log base a (x^y) = y * log base a (x), we can rewrite the expression as follows:
log base 2 (2 ^ (1280 - log base 2 (5)))
Since 2 ^ (1280 - log base 2 (5)) is a power of 2, we can further simplify it to:
2^(1280 - log base 2 (5))
At this point, we cannot simplify it further unless we have specific values for the logarithmic function.
I hope this explanation helps you understand how to solve logarithmic expressions step by step. If you have any further questions, feel free to ask!