Expand 4(7-2x)

[Stephani Kapnick]

Wecan use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:

We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.

Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression [latex]4\mathrmln\left(x\right)[/latex], we identify the factor, 4, as the exponent and the argument, x, as the base and rewrite the product as a logarithm of a power:

[latex]2\mathrmlogx - 4\mathrmlog\left(x+5\right)+\frac1x\mathrmlog\left(3x+5\right)=\mathrmlog\left(x^2\right)-\mathrmlog\left(\left(x+5\right)^4\right)+\mathrmlog\left(\left(3x+5\right)^x^-1\right)[/latex]

[latex]\mathrmlog\left(x^2\right)-\mathrmlog\left(\left(x+5\right)^4\right)+\mathrmlog\left(\left(3x+5\right)^x^-1\right)=\mathrmlog\left(\fracx^2\left(x+5\right)^4\right)+\mathrmlog\left(\left(3x+5\right)^x^-1\right)[/latex]

[latex]\mathrmlog\left(\fracx^2\left(x+5\right)^4\right)+\mathrmlog\left(\left(3x+5\right)^x^-1\right)=\mathrmlog\left(\fracx^2\left(3x+5\right)^x^-1\left(x+5\right)^4\right)[/latex]

[latex]\mathrmlog\fracx^12\left(x+5\right)^4\left(2x+3\right)^4[/latex]; this answer could also be written as [latex]\mathrmlog\left(\fracx^3\left(x+5\right)\left(2x+3\right)\right)^4[/latex].

In chemistry, pH is a measure of how acidic or basic a liquid is. It is essentially a measure of the concentration of hydrogen ions in a solution. The scale for measuring pH is standardized across the world, the scientific community having agreed upon its values and methods for acquiring them.

Suppose C is the original concentration of hydrogen ions and P is the original pH of the liquid. Then [latex]\textP=-\mathrmlog\left(C\right)[/latex]. If the concentration is doubled, the new concentration is 2C. Then the pH of the new liquid is [latex]\textpH=-\mathrmlog\left(2C\right)[/latex]

Two states (South Dakota and North Carolina) implemented Medicaid expansion in 2023, reducing the number of low-income uninsured people nationally without access to Medicaid. Expansion in those two states brings the count to 40 states and the District of Columbia that have adopted expansion, leaving ten states that have not adopted it. Using data from 2022, the most recent year available, this brief presents estimates of the number and characteristics of uninsured people in the ten non-expansion states who could be reached by Medicaid if their states adopted the Medicaid expansion. An overview of the methodology underlying the analysis can be found in the Data and Methods, and more detail is available in the Technical Appendices.

The coverage gap exists in states that have not adopted the ACA Medicaid expansion for adults who are not eligible for Medicaid coverage or subsidies in the Marketplace. The ACA expanded Medicaid to nonelderly adults with income up to 138% FPL ($20,782 annually for an individual in 2024) with enhanced federal matching funds (now at 90%). The Medicaid expansion established a uniform eligibility threshold across states for low-income parents and newly established Medicaid coverage for adults without dependent children. However, the expansion is effectively optional for states because of a 2012 Supreme Court ruling. As of February 2024, 40 states and DC have expanded Medicaid (Figure 1).

States that have not implemented the expansion have uninsured rates that are nearly double the rate of expansion states (14.1% compared to 7.5%). People without insurance coverage have worse access to care than people who are insured. One in five uninsured adults in 2022 went without needed medical care due to cost and uninsured people are less likely than those with insurance to receive preventive care and services for major health conditions and chronic diseases.

Nearly three-quarters of adults in the coverage gap are concentrated in three states in the South. Four in ten people in the coverage gap reside in Texas, which has very limited Medicaid eligibility, and consequently, a large uninsured population (Figure 4). An additional 19% of people in the coverage gap live in Florida and 12% live in Georgia. In total, 97% of those in the coverage gap live in the South. Seven of the 16 states in the South have not adopted the Medicaid expansion, and the region has more low-income, uninsured adults and higher uninsured rates compared to other regions.

People in the coverage gap are disproportionately people of color. Nationally, over six in ten (62%) people in the coverage gap are people of color, a share that is higher than for non-elderly adults generally in non-expansion states (53%) and for non-elderly adults nationwide (46%) (Figure 5). These differences in part explain persisting disparities in health insurance coverage by race/ethnicity.

Despite having low income, nearly six in ten people in the coverage gap are in a family with a worker, and over four in ten are working themselves (Figure 6). Adults who work may still have incomes below poverty because they work low-wage jobs. People with incomes below poverty often do not have access to employer based health insurance or if available, it is often unaffordable. The most common jobs among adults in the coverage gap are cashier, cook, waiter/waitress, construction laborer, maid/housecleaner, retail salesperson, and janitor. For parents in non-expansion states, even part-time work may make them ineligible for Medicaid.

Some people in the coverage gap have significant current health care needs. KFF analysis of the 2022 American Community Survey shows that more than one in six (17%) people in the coverage gap have a functional disability, meaning they have serious difficulty with hearing, vision, cognitive functioning, mobility, self-care, or independent living. Even with functional disabilities, many are not able to qualify for Medicaid through a disability pathway leaving them uninsured. Older adults, age 55-64, an age of increasing health needs, make up 18% of people in the coverage gap. Research has demonstrated that uninsured people in this age range may leave health needs untreated until they become eligible for Medicare at age 65.

If all states adopted the Medicaid expansion, approximately 2.9 million uninsured adults would become newly eligible for Medicaid. This number includes the 1.5 million adults in the coverage gap and an additional 1.4 million uninsured adults with incomes between 100% and 138% FPL, most of whom are currently eligible for Marketplace coverage but not enrolled (Figure 7 and Table 1). Most of the adults who are currently eligible for coverage in the Marketplace qualify for plans with zero premiums; however, even with no premiums, Medicaid could provide more comprehensive benefits and lower cost-sharing compared to Marketplace coverage. The potential number of people who could be reached by Medicaid expansion varies by state.

A substantial body of research continues to point to largely positive effects from the Medicaid expansion. KFF reports published in 2020 and 2021 reviewed more than 600 studies and concluded that expansion is linked to gains in coverage, improvement in access and health, and economic benefits for states and providers. More recent studies generally find positive effects related to more specific outcomes such as improved access to care, treatment and outcomes for cancer, chronic conditions, sexual and reproductive health and behavioral health. Studies also point to evidence of reduced racial disparities in coverage and access, reduced mortality and improvements in economic impacts for providers (particularly rural hospitals) and economic stability for individuals.

Renewed debates over Medicaid expansion could lead to additional states adopting the expansion. Legislative opposition to expansion may be softening in some states, driven by financial challenges facing rural hospitals as well as interest in taking advantage of the additional federal funding. If additional states were to expand Medicaid, it could help limit increases in the number of people who become uninsured because of the unwinding of the Medicaid continuous enrollment provision. In South Dakota and North Carolina, Medicaid enrollment dropped initially after the start of the unwinding, but began increasing again following implementation of Medicaid expansion in each state.

In 2022, New Jersey became the seventh state to enact a refundable state-level Child Tax Credit to help families meet the high costs of raising kids.[i] Under the new program, eligible families will receive up to $500 for every child under six years old as early as January 2023.[ii]

To build on the success of the Child Tax Credit, state lawmakers can and should expand the program with a higher maximum credit and so more families qualify for financial support. This report lays out the logical next steps for the Child Tax Credit and would make New Jersey a more affordable place to start and raise a family.

The age limit would be lifted from children under age 6 to children under age 12. Children do not stop having costs as time goes on.[xi] Expanding the age range will also cover a wider range of families and boost the credit amount for already-eligible families who also have children in the older age range.

3a8082e126