Compare the graphs of the logarithmic functions f(x) =log7 x and g(x)=log4 x. For what values of x is f=g and f<g? Explain how you know.
the bold was to be a subscript
To compare the graphs of the logarithmic functions f(x) = log7 x and g(x) = log4 x, we need to analyze their properties.
1. Graphs of f(x) and g(x):
The function f(x) = log7 x is a logarithmic function with base 7. As x increases, f(x) also increases, but at a decreasing rate. The graph of f(x) will have an asymptote at the y-axis (x = 0) and will approach positive infinity as x approaches infinity.
The function g(x) = log4 x is a logarithmic function with base 4. Similar to f(x), as x increases, g(x) also increases, but at a slower rate than f(x). The graph of g(x) will have an asymptote at the y-axis (x = 0) and will approach positive infinity as x approaches infinity.
2. Values of x when f(x) = g(x):
To find the values of x when f(x) = g(x), we can set the two functions equal to each other:
log7 x = log4 x
Now, we can solve for x using log properties:
log7 x = log4 x
log(x) / log(7) = log(x) / log(4)
Since the logarithms are equal, we can cancel them out from both sides:
1 / log(7) = 1 / log(4)
This equation is not true for any values of x, as it simplifies to 1/log(7) = 1/log(4), which is an invalid equation.
3. Values of x when f(x) < g(x):
To find the values of x when f(x) < g(x), we can compare the rates of increase of the two functions. As mentioned before, f(x) increases at a faster rate than g(x) as x increases.
Therefore, there are no values of x when f(x) < g(x) since f(x) is always greater than g(x) for any positive value of x.
In summary, the graphs of the logarithmic functions f(x) = log7 x and g(x) = log4 x have no points of intersection (f(x) = g(x)) and there are no values of x for which f(x) < g(x). This is because f(x) increases at a faster rate than g(x) for all positive values of x.
To compare the graphs of the logarithmic functions f(x) = log7(x) and g(x) = log4(x), we need to understand the properties and behavior of logarithmic functions.
First, let's establish the basic properties of logarithms:
1. The base of the logarithm determines the scale of the graph. In this case, f(x) has a base of 7 and g(x) has a base of 4.
2. The domain of a logarithmic function is all positive real numbers, which means x > 0.
3. The range of a logarithmic function is all real numbers, which means y can take on any value.
Now, let's examine how the two functions compare:
1. Equality (f = g):
To find the values of x where f(x) = g(x), we need to solve the equation log7(x) = log4(x). We can do this by applying the logarithmic properties:
log7(x) = log4(x)
log(x) / log(7) = log(x) / log(4) (using the change of base formula)
log(x) * log(4) = log(x) * log(7) (cross-multiplication)
log(x) * (log(4) - log(7)) = 0
The only way for this equation to hold true is if either log(x) = 0 or log(4) - log(7) = 0.
log(x) = 0 means x = 1. Therefore, f(x) = g(x) when x = 1.
log(4) - log(7) = 0 means log(4/7) = 0.
Since the logarithm of a number is 0 only when the number is 1, we have 4/7 = 1.
This is not possible, as 4/7 is not equal to 1. Therefore, there are no other values of x where f(x) = g(x).
2. Inequality (f < g):
To determine when f(x) < g(x), we need to compare the values of log7(x) and log4(x).
Since 7 > 4, we can conclude that log7(x) < log4(x) for any positive value of x greater than 1. This means that f(x) < g(x) for all x > 1.
In summary:
- f(x) = g(x) only when x = 1.
- f(x) < g(x) for all x > 1.
By understanding the properties and rules of logarithmic functions, we can determine the equality and inequality of f(x) = log7(x) and g(x) = log4(x) with confidence.
I assume you meant
f(x) = log7 x and g(x) = log4 x
log7 x = log4 x
I know that all log graphs of the form
y = loga x where a > 0 , a not = 1
pass through the point (1,0)
You should know the basic shape of y = loga x
since loga x = logx/loga
and since log7 > log4 , you know which is the larger value for any logx