I am asked to write a logarithmic expresion as a single logarithm with coefficient 1 and simplify as much as possible. The equation is:
4logbase5Y-3logbase5X+1/2logbase5Z.
Could some please help me I am not understanding this whole Logarithm thing.
first of all, it's just an expression, not an equation (no = sign)
Since loga + logb = log(a*b)
just a x^a * a^b = x^(a+b)
log x^n = log(x*x*x*x...)= n*logx
assuming logs base 5, but the base does not matter if all the logs are to the same base; the rules apply regardless of the base
4logx-3logy + 1/2 logz
= log(x^4√z/y^3)
Sure! I'd be happy to help you understand logarithms and simplify the given expression.
To start, let's first review the basics of logarithms. Logarithms are basically the inverse operation of exponentiation. They help us solve equations that involve exponential functions.
In the given expression, we have three terms: 4logbase5Y, -3logbase5X, and 1/2logbase5Z. Our goal is to simplify this expression as much as possible and write it as a single logarithm with coefficient 1.
To do that, we can use the following log rules:
1. Product Rule: log(base b)(MN) = log(base b)M + log(base b)N
2. Quotient Rule: log(base b)(M/N) = log(base b)M - log(base b)N
3. Power Rule: log(base b)(M^p) = p * log(base b)M
Now, let's simplify the given expression step by step:
Step 1: Applying the product rule to the first two terms:
4log(base 5)Y - 3log(base 5)X = log(base 5)(Y^4) - log(base 5)(X^3)
Step 2: Applying the power rule to the third term:
1/2log(base 5)Z = log(base 5)(Z^(1/2))
Step 3: Combining all the terms:
log(base 5)(Y^4) - log(base 5)(X^3) + log(base 5)(Z^(1/2))
Step 4: Applying the quotient rule to combine the first two terms:
log(base 5)((Y^4)/(X^3)) + log(base 5)(Z^(1/2))
Step 5: Simplifying further:
log(base 5)((Y^4 * Z^(1/2))/(X^3))
And there you have it! The given logarithmic expression has been simplified as a single logarithm: log(base 5)((Y^4 * Z^(1/2))/(X^3)).
I hope this explanation helps you understand how to simplify logarithmic expressions. Let me know if you have any further questions!