The given equation is y = log 5 (x + 1) + 1. The term (x + 1) inside the logarithm function indicates a horizontal shift to the left by 1 unit. This means that the graph of the equation will be shifted to the left by 1 unit compared to the standard logarithmic function. Additionally, the term +1 outside the logarithm function indicates a vertical shift upward by 1 unit. This means that the graph of the equation will be shifted upward by 1 unit compared to the standard logarithmic function. Therefore, the correct transformations that apply here are a horizontal shift left 1 and a vertical shift up 1.

Logarithmic functions are the inverse of exponential functions. This means that if we have an exponential function, we can find its corresponding logarithmic function by switching the roles of the base and the exponent. In other words, if y = a^x is an exponential function, then its inverse logarithmic function is given by x = log_a(y).

The graph of the function f(x) = log x is vertically stretched by a factor of 3, meaning that the y-values are multiplied by 3. Additionally, the graph is shifted 1 unit to the left, meaning that all x-values are decreased by 1. Finally, the graph is shifted 5 units down, meaning that all y-values are decreased by 5.

The given equation is y = log 6 (x - 1) - 5. The term (x - 1) inside the logarithm function indicates a horizontal shift right 1 unit, as it shifts the entire graph of the function 1 unit to the right. The term -5 at the end of the equation indicates a vertical shift down 5 units, as it shifts the entire graph of the function 5 units downwards. Therefore, the correct transformations that apply here are a horizontal shift right 1 and a vertical shift down 5.

The logarithm function has a vertical asymptote. This is because as the input approaches negative infinity, the output of the logarithm function approaches negative infinity. Similarly, as the input approaches positive infinity, the output approaches positive infinity. Therefore, the logarithm function has a vertical asymptote at x = 0.

The given expression is -ln(x-2)+3. Since the natural logarithm is only defined for positive numbers, the expression inside the logarithm, (x-2), must be greater than 0. Solving for x, we find that x must be greater than 2. Therefore, the domain of the expression is (2, infinity).

The given function is -ln(x-2)+3. The natural logarithm function ln(x) is defined for all positive real numbers. In this case, the function is -ln(x-2), which means that the argument (x-2) must be positive. Since there are no restrictions on the value of x in the given options, the range of the function is all real numbers.

The given equation is y = -2 log (1/2)x. The coefficient -2 in front of the logarithm function indicates a vertical stretch by a factor of 2. This means that the graph of the equation is stretched vertically, making it appear taller. The negative sign in front of the logarithm function indicates a reflection across the x-axis. This means that the graph is flipped upside down. Therefore, the correct transformations that apply here are a vertical stretch by 2 and a reflection across the x-axis.

The asymptote of the given equation y = log2x - 3 is x = 0. This means that as x approaches 0, the value of y approaches negative infinity. The logarithmic function has a vertical asymptote at x = 0, which means that the graph of the equation will approach but never touch this vertical line.

The given equation is in the form y = 3log3(2x), where the base of the logarithm is 3. This means that there is a vertical stretch by a factor of 3, which is indicated by the coefficient in front of the logarithm. Additionally, there is a horizontal shrink by a factor of 1/2, which is indicated by the coefficient in front of the x. Therefore, the correct transformations that apply here are a vertical stretch by 3 and a horizontal shrink by 1/2.